Linking forest growth with stand structure: Tree size inequality, tree growth or resource partitioning and the asymmetry of competition

Forest Ecology and Management 447 (2019) 139–157

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Tamm reviews

Linking forest growth with stand structure: Tree size inequality, tree growth or resource partitioning and the asymmetry of competition

T

David I. Forrester Swiss Federal Institute of Forest, Snow and Landscape Research WSL, Zürcherstrasse 111, 8903 Birmensdorf, Switzerland

ARTICLE INFO

ABSTRACT

Keywords: Asymmetric competition Climate change Continuous cover forestry Growth dominance Mixed species Multi-aged Size heterogeneity Uneven-aged

Stand structure can strongly influence forest growth and other processes, such as the water balance, carbon partitioning, nutrient cycling and light dynamics. However, individual structural variables can be positively or negatively correlated with growth. This is the case for variables such as size inequality and those that describe resource partitioning, such as the degree of symmetric/asymmetric competition and growth dominance. Several contrasting growth-structure correlations are reviewed and linked to forest processes by considering the different types of tree interactions they are associated with. Contrasting growth-structure correlations appear to converge when they are examined using a simple framework where stand growth is a function of three variables as opposed to any one of the variables alone; stand density, size distributions and tree size-growth relationships. The size distributions quantify how the stand density is distributed between the different sizes while the sizegrowth relationships quantify how growth is partitioned between different sizes. Size inequality may not often be a useful explanatory variable and instead it appears to sometimes correlate with growth because it can be correlated with other variables that influence growth. The spatial and temporal dynamics of the effects of structure on growth have received little attention and a long-term growth and yield data set from central Europe was used to examine how the effects of structure can change along climatic gradients. The simple framework of three variables could be used to separate the effects of structure and functioning when comparing mixed and monospecific forests, as well as to design silvicultural interventions or to determine whether past management interventions have achieved their goals. The implications for selecting which structural variables to use and when scaling up to the stand level, are also discussed.

1. Introduction Many forest processes (e.g. water balance, nutrient cycling, carbon partitioning, light absorption) are influenced by stand structure and vice versa (Pretzsch et al., 2015). The importance of structure on forest processes and the ease with which it can be manipulated by management highlights the value of understanding the relationship between structure and growth and how this relationship is influenced by climatic and edaphic conditions, species mixing and silviculture. For example, thinning has been proposed for mitigating the effects of drought in the short term of years or decades (Sohn et al., 2016) and species mixing has been suggested for reducing drought susceptibility in the longer term (Grossiord et al., 2014), however the effects of thinning and species mixing on growth can strongly depend on the stand structure (del Río and Sterba, 2009; Forrester et al., 2013b; Forrester, 2014) which can therefore influence, and possibly even reverse, their effect. Relationships between stand growth and a given structural variable, and whether the correlations are positive or negative, can vary greatly

between forest types and species. For example, depending on the context, stand growth can increase, decrease or not change with growth dominance, which quantifies how the stand growth is partitioned between different sized trees within the stand (Section 3). Similarly, depending on the context, stand growth increases, decreases or does not change with increases in size inequality (Section 5), where size inequality is often quantified as the Gini coefficient (Weiner and Solbrig, 1984), coefficient of variation or the Shannon diversity index (Shannon, 1948) calculated from tree size. This diversity in the directions of stand growth-structure relationships suggests that there could be a missing link that needs to be considered before a convergence of these results is realised. The main objective of this review was to provide a simple framework that can explain these contrasting results, and that can be used when designing sampling approaches and silvicultural treatments. One of the most important stand structural characteristics that determines growth is stand density, such as the number of trees, stand basal area, leaf area index and stand density index. For a given age and site, growth tends to increase with stand density before almost

E-mail address: [email protected]. https://doi.org/10.1016/j.foreco.2019.05.053 Received 26 March 2019; Received in revised form 19 May 2019; Accepted 21 May 2019 Available online 30 May 2019 0378-1127/ © 2019 Elsevier B.V. All rights reserved.

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Fig. 1. An illustration of Eq. (1). The frequency distributions are the red curves in the left panels. These are the products of the stand density and the probability density functions of Eq. (1). The individual tree size-growth relationships are blue curves in the upper panels. All other panels (with black curves) are the result of the given frequency distribution × size-growth relationship, and indicate the resulting stand basal area growth (Gr; m2 ha−1 year−1) and the growth dominance coefficient (DCgrowth) with its associated dominance curve (solid black line). The black dashed lines are 1:1 lines. The frequency distribution and size-growth relationships are in terms of individual tree basal area (cm2) and basal area growth (cm2 year−1). All stands have a stand density of 40 m2 ha−1. This graph is based on growth but it could be based on processes like transpiration, APAR, drought stress etc. While some of the size-growth shapes are unlikely, they are included to facilitate the illustration of the contrasting shapes. For simplicity, no complex size distributions are shown (e.g. bi-modal).

plateauing after the leaf area has peaked (Langsæter, 1941; Zeide, 2004). Other forest processes also change with stand density, such as transpiration (Bréda et al., 1995; Alsheimer et al., 1998; Forrester et al., 2012) and the absorption of photosynthetically active radiation (APAR) (Wang and Jarvis, 1990b; Marková et al., 2011; Binkley et al., 2013). Size distributions are another structural characteristic that has a strong influence on productivity and is often used to quantify stand structure (Vanclay, 1994; Weiskittel et al., 2011). Size distributions describe how the stand density, e.g. stand basal area, leaf area, volume, is distributed between size classes within the stand. For a given density, productivity varies greatly with size distributions. Productivity will be lower when the size classes that make up a higher proportion of the stand have low growth rates (Fig. 1). Therefore, a third defining component of structure–productivity relationships are relationships between individual tree size and individual tree growth (size-growth relationships). The size-growth relationships quantify how growth differs between size classes and how the stand density and size distributions will change through time and following mortality and recruitment (Westoby, 1982). As a result, they are sometimes even called distribution-modifying functions (Westoby, 1982). Since productivity is so strongly influenced by density, size distributions and size-growth relationships, these characteristics are often directly modified to achieve many management objectives and to specifically define silvicultural systems (Assmann, 1970; Pukkala and Gadow, 2012; Messier et al., 2013; O'Hara, 2014; Pretzsch et al., 2017). They are also often foundational components of forest growth models (Vanclay, 1994; Weiskittel et al., 2011). The effects of stand structure on growth can be described and quantified using Eq. (1) as a framework.

Stand growth = stand density × size probability density function × size relationship

growth (1)

where size and density are based on the same variable and tree and stand growth are based on the same variable. For example, stand basal area growth (m2 ha−1 year−1), stand density in terms of basal area (m2 ha−1), size distribution in terms of a probability distribution based on tree basal area classes (m2) and relationships between individual tree basal area (m2 tree−1) and basal area growth (m2 tree−1 year−1). Eq. (1) also applies to other characteristics when stand and tree growth are replaced with variables relating to light absorption, drought stress, transpiration and resource-use efficiency (RUE). The size distributions in Eq. (1) can also quantify the vertical distribution of trees due to the relationship between height and size. Since the vertical and horizontal structure of a forest influences, and is influenced by, the resource availability and climatic conditions within the stand (Pretzsch et al., 2015), there is a direct feed-back between all three variables on the right side of Eq (1) (Fig. 2). For example, with a higher stand density, there is likely to be a size-growth relationship with a lower yintercept and/or slope. Size-growth relationships are shaped by how trees interact with each other and therefore provide a link between the effects of structure and processes on growth. They depend on tree interactions, resource availability, RUE and climatic conditions (Section 3). Spatial structure, such as the clustering of trees, can also influence stand growth (Section 5), and this could be examined using Eq. (1) after dividing the stand into individual tree neighbourhoods of different densities and examining the variables in Eq. (1) in each density, as described in Section 140

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Fig. 2. The main components of the relationship between stand growth and stand structure, based on Equation (1). The sizegrowth relationships determine how the size distributions change (see also Fig. 9). The size distributions feed-back to influence size-growth relationships via their effects on within-stand climatic conditions and resource availability. The size-growth relationships are a function of size-uptake and size-RUE relationships (Section 3) which depend on resource availability and climatic conditions. Management and other disturbances directly influence the stand density, size distributions and stand micro-climate and resource availability.

5. Mortality and recruitment can be considered in terms of their effects on all three explanatory variables in Eq. (1). Fig. 1 shows the very large effects that size distribution shapes and size-growth relationships have on stand growth rates (see Gr in Fig. 1), even for the same stand density, and therefore the potential effect of management interventions that can manipulate these characteristics. This illustrates the effect of the relationship described by Eq. (1). That is, stand growth will increase (for a given stand density) if the sizegrowth relationship changes in favour of the tree sizes that represent a larger proportion of the stand density. If the growth rates increase only for sizes contributing a small proportion of the stand, there will only be a minor change in stand growth. For example, when a linear sizegrowth relationship changes from having a y-intercept of 0 (Fig. 1a, 1i, 1q, 1y) to a negative y-intercept (Fig. 1b, 1j, 1r, 1z), the stand growth is reduced for all size distribution shapes (for a given density) because growth is lower for all tree sizes. This reduction in growth will be greater if the size-growth curve is convex shaped ( Fig. 1h, 1p, 1x & 1af). The reduction in growth is not as severe for the left-skewed distribution because there is a higher proportion of stand density composed of fast growing large trees (Fig. 1). Similarly, for a given sizegrowth relationship and stand density, stand growth will be highest for distributions with more stand density in the size classes with the highest growth rates. This basic framework could potentially be used to explain the contrasting relationships between stand growth and stand structural characteristics such as tree size inequality (Section 5) or growth dominance and the degree of asymmetric/symmetric competition, which describe the partitioning of growth and resource partitioning between individual trees (Section 3, Fig. 3). Eq. (1) indicates three variables that need to be considered instead of reducing structure to a single more complex variable (e.g. size inequality). A similar analogy for the value of using more than one variable is Beer’s Law, which predicts how much light penetrates plant canopies as a function of the leaf area index and

an extinction coefficient (Monsi and Saeki, 2005). Neither one of these variables alone is adequate for describing light penetration, but together they can be used to describe light penetration for plant canopies in general. The first objective of this review was to identify types of interactions that can occur between trees of different sizes and therefore how growth and resources are partitioned between individuals of different sizes within a stand. This included a review of how size-growth, sizeuptake or size-RUE relationships (also symmetry/asymmetry of competition) are influenced by age, stand structure, stand density, climate and edaphic characteristics. The second objective was to examine whether Eq. (1) can be used to explain why contrasting growth-stand structure relationships are observed when stand growth is described using only a single variable (e.g. growth dominance or tree size inequality). Thirdly, this information was used to examine how the effects of stand structure can change with resource availability, climatic conditions, mixing species, age and management. Finally, the fourth objective was to consider the implications of Eq. (1) with regards to sampling, developing or assessing silvicultural regimes and for splitting structural and functional effects of species mixing on growth. 2. Data and analyses from central European forests Examples from the literature have been used when possible. However, there are several patterns and relationships that have rarely been examined. Therefore, a few relationships between growth and structure or light partitioning were examined using data from Swiss forests where long time series (> 100 years) are available. These data come from the Experimental Forest Management (EFM) plots, which are used to examine silvicultural treatments across a range of species, climate and edaphic conditions (Forrester et al., 2019). The silvicultural treatments include even-aged plantations to single-tree selection forests (e.g. plenter forests), pruning, thinning, provenance, regeneration and 141

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Fig. 3. An example of the size-growth relationships and size distributions associated with negative (a,c,e) or positive (b,d,f) correlations between stand growth and DCgrowth. A set of Larix decidua plots are shown in the left column (a,d,e). As the stands developed, the DCgrowth increased because larger trees grew more relative to their size, as shown by the positive convex size-growth relationship (e). Despite this, the stand growth still declined with age because the large, fast-growing trees contributed only a small amount of stand density, as shown by the reverse-J or right-skewed size distributions (c). In contrast, DCgrowth increased with the growth of a population of regenerating Fagus sylvatica trees within a mixed single-tree selection forest (with Abies alba and Picea abies) (b). The DCgrowth was always negative, but increased with time, mainly because the slope of the size-growth relationships became less concave. The statistics for the size distributions and size-growth relationships are shown in Appendix B.

spacing trials. The analyses in this study used plots located at altitudes between 217 and 1950 m. Mean daily minimum temperature in January ranged from −10.4 °C to −0.3 °C and the mean daily maximum temperature in July ranged from 12.2 °C to 25.1 (1970 to 2014). Mean annual precipitation was between 692 and 2331 mm (1970 to 2014). A total of 1344 plots were used (746 sites) and the mean plot area was 0.25 ha (0.005–3.1 ha).

area were predicted using species-specific relationships that incorporate the effects of stand basal area (Forrester et al., 2017). The dominance coefficient was calculated using Eq. (4) (Section 3). 2.2. Calculation of APAR Individual-tree absorption of photosynthetically active radiation, APAR (GJ tree−1 year−1), was calculated using the 3D tree-level model Maestra (Grace et al., 1987; Wang and Jarvis, 1990a; Medlyn, 2004; Duursma and Medlyn, 2012), which has been validated in several mixed and monospecific forests (Wang and Jarvis, 1990a; Charbonnier et al., 2013; le Maire et al., 2013; Forrester et al., 2018). The APAR is predicted from the crown architecture (crown width and length, leaf area and leaf angle distributions), species-specific differences in leaf optical properties and leaf area density distributions. Shading by neighbouring trees is calculated by representing the canopy as an array

2.1. Tree measurements and calculations For all trees, the diameter at a height of 1.3 m (DBH) was measured, while for the height, four crown radii and height to the lowest maincrown branch, only a sample of trees (the 100 largest-diameter trees and 20% of the rest) were measured and these variables were then predicted for all other trees using plot-, year- and species-specific regressions (Forrester et al., 2019). Individual tree stem mass and leaf 142

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Table 1 Processes that can influence the interactions between trees of different sizes. The majority of these processes are likely to interact with others. Name of process

Light-related Canopy stratification

Size classes likely to benefit*

Effect on production ecology (see Section 3, Eq. (3))

References

Large

Crowns higher up in the canopy experience higher light availability and may absorb more light for a given crown architecture and leaf area. Size related (or light availability related) differences in crown architecture that increase light absorption per unit leaf area and enable trees to grow under different light conditions. Crown architectural differences can include the within-crown vertical distributions of leaf area, leaf angles, specific leaf area, leaf-area density and differences in crown shapes or allometry (e.g. stem diameter to leaf area, crown width, crown length etc). The vertical light gradient in forest canopies could influence or interact with the physiology of trees of different sizes or ages. These may include size-related differences in photosynthetic rates, LUE, shade tolerance etc. Such changes in physiology could facilitate the maintenance of the canopy structures described above. However, leaves exposed to higher light intensity may also suffer more from stresses associated with heat, water and photoinhibition. Younger or smaller individuals of deciduous species may produce foliage earlier and defoliate later than taller or older and more well-lit individuals, thereby extending the growing season. This may result mainly from ontogenic differences and to a lesser degree from more favourable environmental conditions in the lower canopy (e.g. warmer, protection from frost).

(Gersonde et al., 2004; Forrester et al., 2018)

Different crown shapes and architectures

All sizes

Physiological differences

All sizes

Phenological differences

Small

Water-related Spatial stratification and hydraulic redistribution

Large or all sizes

Hydraulic architecture, tree allometry

Small

Shared mycorrhizal networks

All sizes

Canopy interception

Large

Transpiration and water-use efficiency

Large?

Litter layer as a sponge or barrier

Large or all sizes

Modified within-stand environmental conditions

Small

Larger trees can have deeper root systems allowing access to deeper soil water, which can make them less sensitive to shortterm drought events. They may also extend further into canopy gaps. Shallow rooted trees could access water quickly after rainfall events before it evaporates or seeps deeper into the soil profile. Deeper rooting trees may lift water from deep moist layers to shallower drier layers. Taller trees with longer branches can have higher resistance to water flow. They may also experience greater evaporative demands higher up in the canopy when irradiance, temperature and VPD are higher. This may sometimes cause greater water stress of larger trees. Partitioning to roots can decline with increasing age and size, which may also result in size-related differences in responses to water availability and climatic conditions. Carbon, water, and nutrient uptake may increase if these resources are transferred via a shared root and mycorrhizal network with other trees. Taller trees and trees with larger crowns can intercept precipitation and funnel it down their stems towards their roots, thereby increasing water availability for their roots and reducing the proportion of precipitation available to other trees. Intra-specific differences (e.g. due to size and age) in water-use efficiency or transpiration rates (per unit crown projection area or sapwood area or diameter) will influence whether water availability (and uptake) is higher or lower for neighbouring trees. Litterfall decreases exponentially with distance from the tree, and may result in more litter under trees with larger crowns. Similarly, individual tree litterfall can increase exponentially with tree stem circumference. Water storage may increase if the O horizon becomes deeper (increased water availability and uptake). In contrast, infiltration of precipitation into the O horizon could be reduced causing higher evaporation and runoff (reduced availability and uptake). Smaller trees may be facilitated by changes in the vertical environmental gradient within the canopy caused by the transpiration of taller trees reducing the vapour pressure deficit, temperature and by shading.

(Niinemets and Kull, 1995; Maguire and Bennett, 1996; Niinemets, 1996; Niinemets, 1997; Gspaltl et al., 2013; Sapijanskas et al., 2014; Guisasola et al., 2015; Dong et al., 2016; Forrester et al., 2018)

(O'Hara, 1996; Grulke and Retzlaff, 2001; Niinemets, 2002; Niinemets and Valladares, 2004; O'Hara and Nagel, 2006; O'Grady et al., 2008)

(Augspurger and Bartlett, 2003; Lopez et al., 2008; Vitasse, 2013; Gressler et al., 2015)

(Orwig and Abrams, 1997; Liu and Muller, 1993; Vose and Swank, 1994; Biondi, 1996; Dawson, 1996; Burgess et al., 1998; Grulke and Retzlaff, 2001; Wichmann, 2001; Drake et al., 2009; Niinemets, 2010a; Christina et al., 2011; Neumann and Cardon, 2012; Prieto et al., 2012; Pinheiro et al., in press) (Grulke and Retzlaff, 2001; Ryan et al., 2006; Litton et al., 2007; Niinemets, 2010a; Bennett et al., 2015; McDowell and Allen, 2015)

(Simard et al., 2012) (Aston, 1979; Crockford and Richardson, 2000; Schume et al., 2004; André et al., 2008; Augusto et al., 2015)

(Meinzer et al., 2005; Kunert et al., 2012; Otto et al., 2014; Forrester, 2015)

(Schume et al., 2004; Jonard et al., 2006; Ilek et al., 2015)

(Liu and Muller, 1993; Niinemets and Valladares, 2004; Grote et al., 2016)

(continued on next page)

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Table 1 (continued) Name of process

Size classes likely to benefit*

Effect on production ecology (see Section 3, Eq. (3))

References

Nutrient-related Nutrient cycling

Large

(Staelens et al., 2004; Jonard et al., 2006; Gómez-Aparicio and Canham, 2008; Boyden et al., 2012; Uriarte et al., 2015)

Foliar nutrition

All?

Chemical, spatial or temporal stratification

All sizes

Since litterfall decreases exponentially with distance from the tree and litterfall (per tree) can increase exponentially with tree size, there could be spatial variability in the litter layer. This could result in accelerated rates of nutrient cycling due to greater nutrient contents (and concentrations) of litterfall and decomposition rates, resulting in greater nutrient availability and uptake. Foliar nutrition can vary with tree size due to age or position within the canopy and influence gas exchange. Age- and size-specific differences in mycorrhizal associations, fine-root architecture, vertical distributions of roots or seasonality in root growth and resource uptake could result in reductions in competition for nutrients and hence greater total resource uptake.

Factors that potentially influence multiple resources Carbon partitioning All sizes Carbon partitioning and tree allometry influence the ability of trees to acquire and use resources and can change in response to competition or stand structures that influence resource availability, such as changes in stand density. Partitioning could also change when different sizes have access to more resources than others. Insect herbivory and leaf All sizes Susceptibility to attack and responses to attack can differ pathogens depending on tree size or the size of neighbours. Mortality All sizes Many causes of tree death are size dependent, but the sizes that are most susceptible depend on the cause.

(Niinemets, 1997; Grulke and Retzlaff, 2001) (Gardner and Malajczuk, 1988; Bellei et al., 1992; Pinheiro et al., in press)

(Mokany et al., 2006; Litton et al., 2007; Poorter et al., 2012; Grote et al., 2016)

(Newton and Jolliffe, 1998; Barbosa et al., 2009; Niinemets, 2010a) (Niinemets, 2010a; Holzwarth et al., 2013; Brang et al., 2014; Bennett et al., 2015; McDowell and Allen, 2015; Grote et al., 2016)

* The “All sizes” indicates that any size could benefit from this interaction, depending on the context.

of tree crowns with positions defined by x and y coordinates such that the slope and aspect are considered in both the x and y directions. Parameter values were obtained from the literature (Appendix A). The APAR of evergreen species was total annual APAR, while for deciduous species, growing season APAR was used where the growing season for each species was calculated as a function of altitude (Dittmar and Elling, 2006; Vitasse et al., 2009; Čufar et al., 2012; Pellerin et al., 2012; Cornelius et al., 2013; Schuster et al., 2014). Edge effects on APAR calculations were minimised by excluding trees from analyses that were within 10 m of plot boundaries and by simulating an additional 20-m-wide buffer around each plot based on the average tree spacing, species composition and tree dimensions of the given plot.

Core Team, 2018). All fixed effect variables were initially included before non-significant (P > 0.05) variables were removed in order of decreasing P-value. To ensure that residuals were centred at zero and approximately normally distributed, the residual and normal quantile plots were assessed. For tree-level analyses, the random effect was tree nested within plot, and for stand-level analyses, the random effect was year nested within plot. 3. Production ecology of tree interactions The production ecology equation is a useful tool to quantify the effects of stand structure, and its modification through management, on productivity (Eq. (3); Monteith, 1977; Binkley et al., 2004).

2.3. Climatic data

Net primary production

Climatic data, including solar radiation data for the Maestra APAR calculations, was collected by the Federal Office of Meteorology and Climatology MeteoSwiss (Bundesamt für Meteorologie und Klimatologie MeteoSchweiz) and was interpolated for each plot location using the methods described by Thornton et al. (1997). Topographic variables, including altitude, were obtained using the Land suitability models from the Swiss Federal Statistical Office.

= resource supply × fraction of resource acquired × resource efficiency

Linear mixed models were used to examine relationships between variables in the form of Eq. (2). (2)

ij

(0, (0, With random effects where yij is the dependent variable, Xij is the independent variables matrix (fixed effects matrix), Zij is the random effects matrix, β and b are the parameters’ vector of fixed and random effects, i is the index for plot, j is the index for year or tree and εij is the within group error vector. Mixed models (Eq. (2)) were fitted as hierarchical mixed-effects models using the R package nlme (Pinheiro et al., 2018) and R 3.5.1 (R biiid N

2 i );

bijiid N

(3)

Based on the production ecology equation, increases in growth can result from increases in resource availability, resource uptake and/or RUE. This equation has been used to understand and quantify growth responses to site, climate, age, silviculture, species mixing and stand density (Richards et al., 2010; Binkley, 2012; Forrester, 2013; 2014). In the context of growth-structure relationships, it is necessary to consider the production ecology equation at both the tree and stand levels. At the stand level, changes in size distributions, can cause increases or decreases in stand growth even if there are no changes at the tree level, in terms of sizegrowth, size-uptake or size-RUE relationships (Eq. (1), vertical columns in Fig. 1). At the individual tree level, resource uptake and RUE both depend on the climate and edaphic conditions within the stand and how individual trees interact with each other (Table 1). Therefore there is a direct feedback between the tree interactions and the stand structure because the structure (e.g. density, size distributions) influences the micro-climate and resource distributions within the stand, and these influence the resource uptake and RUE of individual trees, which influences their growth and hence the stand structure (Fig. 2).

2.4. Linear regression analyses

yij = Xij + Zij bij +

use

2 ij )

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3.1. Growth dominance and the partitioning of growth and resources between trees

one increases without any decline in the other. This is a common response to increased resource availability, improved climate conditions and improved RUE at the whole stand level, such as in response to fertiliser application, site quality, intermediate ages, thinning from below and species mixing (Table 1) (West and Osler, 1995; Pretzsch and Biber, 2010; Forrester et al., 2011; Campoe et al., 2013a; Forrester and Albrecht, 2014; Vallet and Perot, 2018). Although non-normalised sizegrowth relationships also indicate the partitioning between individual trees (i.e. the shape of the curve or line), this partitioning effect is confounded with any total stand growth increase or decrease effect. In contrast, the use of normalised data to calculate DC and SGR, isolates the partitioning effect. This partitioning indicates which tree sizes benefit, at the expense of others. Fig. 1 indicates how changes in partitioning could influence productivity for different size distributions, without any change in stand density. Lastly, the DC and SGR vary depending on the size distribution (Fig. 1) and should therefore be calculated from all trees within the stand or from a random sample (i.e. each size sampled at a frequency proportional to its abundance). In contrast, the non-normalised sizegrowth equations do not depend on the size distribution and therefore only show the correlation between size and growth. This means they do not show partitioning per se, because while partitioning varies with the size distributions, the size-growth relationships do not.

Tree interactions can be examined directly by comparing individual tree growth, resource uptake and RUE (Table 1). However, to link individual tree interactions with stand structure, it is pertinent to summarise tree interactions on a whole population level by examining how the stand growth or resources are partitioned between trees within the stand. This indicates which trees, relative to their size, contribute the most growth, are the most competitive at acquiring resources or the most resource-use efficient. That is, while partitioning of growth is not related to stand productivity per se, if partitioning changes in favour of the tree sizes that are abundant within the size distribution, then the productivity will increase, or vice versa (Fig. 1). Therefore, stand growth can be positively or negatively correlated with variables that quantify partitioning (Fig. 3), such as the dominance coefficient or sizegrowth relationships (Section 3.3). Partitioning is included in Eq. (1) in the form of the size-growth/ uptake/RUE relationship. However, more than one variable (e.g. an intercept and a slope) is typically required to describe this relationship. An efficient alternative, based on the same information, is the dominance coefficient (Binkley et al., 2003a; Binkley et al., 2003b; Binkley, 2004). This is often illustrated in the form of dominance curves, as shown in Fig. 1. A dominance curve is calculated by ranking the trees in increasing order of size (e.g. individual tree basal area) and calculating the cumulative proportion of the stand basal area along the x-axis. The y-axis is the cumulative proportion of stand basal area growth, with trees ranked in the same order as the x-axis. The dominance coefficient (DC) can be calculated using Eq. (4) (West, 2014).

3.3. Factors that influence growth partitioning The growth of even-aged stands tends to increase to a maximum rate after the canopy closes and then begins to decline (Mar:Möller, 1947; Ryan et al., 1997). The DC has been used to test hypotheses about whether changes in growth partitioning could be partly responsible for this pattern (Binkley, 2004; Binkley et al., 2006). Four DC phases have been proposed, starting with an open-growing phase, where trees compete only weakly and the growth of each tree is proportional to the fraction of mass it contributes to the stand (1:1 lines in Fig. 1, DCgrowth = 0). During the second phase, the largest trees become relatively more dominant, the DCgrowth increases and the size distribution becomes broader (DCgrowth > 0). The third phase begins as the relative dominance of the larger trees begins to decline until the growth of each tree is proportional to the fraction of mass that it contributes to the stand (1:1 line, DCgrowth = 0). During the final phase, the smaller trees contribute proportionally more growth than they do to stand biomass or basal area etc (DCgrowth < 0). Data sets do not often cover all four phases, but many have shown changes in DCgrowth that are consistent with this temporal pattern, at least for the phases they do cover (Binkley et al., 2003a; Binkley et al., 2003b; Binkley et al., 2006; Tschieder et al., 2012; Pommerening et al., 2016; Baret et al., 2017; Pothier, 2017; Soares et al., 2017). In contrast, there have also been temporal patterns inconsistent with the pattern described above. For example, some have found that DCgrowth was close to 0 or negative, sometimes for a wide range of ages (Binkley et al., 2006; Binkley and Kashian, 2015; Looney et al., 2018), indicating that the large trees never became relatively more dominant than the smaller trees. Or the opposite, where Eucalyptus plantations stayed in the second phase where larger trees were relatively more dominant (Doi et al., 2010), although this may eventually change because eucalypts can live for several centuries and this stand was about 70 years old. Partitioning also changes as resource availability or climatic conditions vary spatially due to natural gradients or treatments. The relative dominance of larger trees increased along site productivity gradients in central Europe (Pretzsch and Biber, 2010; Pretzsch and Dieler, 2011) but declined as stand productivity increased in the eastern states of the USA (Dye et al., 2019). Mixing Eucalyptus saligna with a nitrogenfixing species, Falcataria moluccana, reduced the relative dominance of larger trees by influencing both individual tree growth and mortality, compared with monocultures of either species (Binkley et al., 2003b). Thinning can also increase or decrease DC. Thinning can firstly

n

DC = 1

(si

si

1 )( i

+

i 1)

i=1

(4)

where si is the cumulative proportional size of tree i, and Δi is the cumulative proportional growth (or resource uptake or RUE etc) of tree i. A very similar coefficient, the “Size-Growth Relationship” (SGR) is the slope of a linear relationship between the proportional stand growth (individual-tree growth per stand growth) as a function of the proportional size (individual tree basal area per stand basal area) (Metsaranta and Lieffers, 2010; Castagneri et al., 2012; Looney et al., 2018; Dye et al., 2019). The DC can be between −1 and 1, with DC = 0 indicating that trees grew in proportion to their size (neutral dominance, curve along the 1:1 line, SGR = 1), DC > 0 indicating that large trees grew disproportionately more for their size (positive dominance; curve below the 1:1 line, SGR > 1), and DC < 0 indicating that small trees grew disproportionately more for their size (negative dominance; curve above the 1:1 line, SGR < 1) (Binkley, 2004). 3.2. Implications of normalised vs. non-normalised calculations of partitioning It is critical to distinguish between normalised vs. non-normalised calculations of partitioning because each provides different information. The absolute size-growth/uptake/RUE relationships used in Eq. (1) (Fig. 1), are based on raw data, rather than normalised data such as the SGR, and therefore combine two types of responses: growth partitioning and actual growth. These actual size-growth relationships are required for Eq. (1), but do not isolate the effect of partitioning. In contrast, the DC and SGR both specifically only show the partitioning of growth (or uptake or RUE) within the stand (but are not used in Eq. (1)) because they are based on normalised data, where the growth or size of each individual is expressed relative to the whole stand. The non-normalised size-growth relationships (in Eq. (1)) can indicate an overall increase (or decrease) in stand growth, such as when all trees grow faster, and the slope and intercept both increase, or only 145

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Fig. 4. The relationship between the partitioning of stem mass growth (kg tree−1 year−1), between individual trees in a stand, and the partitioning of absorbed photosynthetically active radiation (APAR in GJ tree−1 year−1; a-f) or light-use efficiency (LUE as growth per APAR; g-l) for six species in central Europe using the data described in Section 2. Partitioning is quantified using the dominance coefficient (Eq. (4)) calculated from individual tree stem growth (DCgrowth), APAR (DCAPAR) or LUE (DCLUE). The DCAPAR and DCLUE are calculated by quantifying tree size in terms of either stem mass (black points) or leaf area (orange points). The Larix data include L. decidua and L. kaempferi.

instantaneously change DC by removing trees from specific size classes, and secondly, in the longer term, thinning can change DC via the change in density and corresponding changes in competition or RUE (Soares et al., 2017). Thinning by removing smaller trees, or all sizes, in Pinus resinosa stands resulted in all sizes having a similar relative dominance (DCgrowth = 0), compared with unthinned stands, where larger trees were relatively more dominant (DCgrowth > 0) (Bradford et al., 2010). Thinning by removing only the larger trees resulted in a higher relative dominance of smaller trees (DCgrowth < 0) at low stand densities and larger trees a higher densities (DCgrowth > 0) (Bradford et al., 2010). Similarly, while larger trees were relatively more dominant in unthinned Pinus taeda stands (DCgrowth > 0), thinning by removing smaller trees resulted in stands where all sizes had a similar relative dominance (DCgrowth = 0) (Tschieder et al., 2012). At higher stand densities in Picea abies stands, all classes had a similar relative dominance (SGR approached 1), while smaller trees were relatively more dominant in less dense plots (Castagneri et al., 2012). Most studies did not examine resource uptake or RUE, but based on the production ecology equation (Eq. (3)) the changes in DC are likely to reflect the changes in uptake or RUE (Section 3.4). Table 1 indicates which size classes might benefit from different factors. When large trees benefit, DC will increase, while if small trees benefit, DC will decrease. Growth partitioning can also change temporally due to inter-annual climatic variability. Several studies found that larger trees grew relatively more as stand growth rates increased (Wichmann, 2001; Metsaranta and Lieffers, 2010; Pretzsch and Dieler, 2011) while no direct effects of inter-annual climatic variability were found in Picea abies forests (Castagneri et al., 2012). It is worth noting that stand growth can be positively, negatively or not correlated with DCgrowth, SGR or slopes of size-growth relationships (Metsaranta and Lieffers, 2010; Coomes et al., 2011; Pretzsch and Dieler, 2011; Dye et al., 2019) and which of these occurs depends on whether the change in the DCgrowth favours the size classes contributing significantly to the stand density (Figs. 1–3). For example, negative correlations where stand growth declines as DCgrowth increases could be expected for bell-shaped and reverse-J shaped size distributions when large trees contribute too little to stand density. The types of interactions that could increase the dominance of larger trees are indicated in Table 1.

resources and size-related differences in RUE. Size-uptake relationships indicate how competition influences resource uptake, in relation to size. That is, they quantify the partitioning of a resource between individuals in a stand. Competition for resources may theoretically be anywhere along a continuum from absolute symmetric to absolute asymmetric competition (Schwinning and Weiner, 1998; Rasmussen and Weiner, 2017; Fernández-Tschieder and Binkley, 2018). The contrast between symmetric and asymmetric interactions is also referred to as the mode of competition (Hara, 1993). These studies define many shapes of sizeuptake relationships to indicate a range of symmetric to asymmetric competition relationships, with some shapes similar to the size-growth shapes in Fig. 1. An alternative quantification of resource partitioning is to calculate DC (or SGR) using Eq. (4). This simplifies the many different curve possibilities into a single number. Therefore, whenever possible DCuptake or DCRUE will be used below. Table 1 lists many types of interactions that occur between trees of different sizes and indicates which interactions favour which size classes. The interpretation of competition symmetry and asymmetry patterns has been complicated by contrasting definitions and assumptions. Three of these are described below because they have critical implications for interpreting tree interactions and the effects of stand structure on productivity. Firstly, regardless of whether growth, uptake or RUE is of interest, the choice of size variable is critical because its relationship to the growth, uptake or RUE determines the value of the DC (Weiner and Thomas, 1992; Ex and Smith, 2014; Looney et al., 2018). For example, size variables like leaf area are often linearly correlated with APAR (Binkley et al., 2013), but size variables like diameter could still be nonlinearly correlated with APAR, in the same stand, because diameter and leaf area are often non-linearly correlated (Forrester et al., 2017). In this case, diameter-APAR relationships will indicate size-asymmetric competition (DCAPAR > 0) while leaf area-APAR relationships could indicate size-symmetric competition (DCAPAR = 0). There are similarly contrasting correlations between diameter, sapwood area and transpiration (Meinzer et al., 2005). The effects of using stem mass to quantify size, as opposed to leaf area or sapwood area, are illustrated in Figs. 4 and 5. The most appropriate size variables for quantifying competition for any given resource may be those involved in its acquisition. For example, leaf area could quantify size when examining competition for light and sapwood area could quantify size when examining competition for water (Figs. 4 and 5). Regardless of the variable that is used, it is important to be clear about the implications of the size variable on defining the DC or size-uptake relationships. Secondly, it is often assumed that aboveground competition is size-

3.4. Resource partitioning and the symmetry and asymmetry of competition As indicated by the production ecology equation (Eq. (3)), changes in growth partitioning could result from changes in the partitioning of 146

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Fig. 5. The relationship between the partitioning of stem mass growth (kg tree−1 year−1) and the partitioning of photosynthetically active radiation (APAR in GJ tree−1 year−1; a), transpiration (litres tree−1 per year−1; b), light-use efficiency (LUE as growth per APAR; c) and water-use efficiency (WUE as growth per transpiration; d) for 24 Eucalyptus nitens plots in a thinning × pruning × fertiliser application experiment near Carrajung, south-eastern Australia. Partitioning is quantified using the dominance coefficient (Eq. (4)) calculated from individual tree stem growth (DCgrowth), transpiration, (DCtranspiration), APAR (DCAPAR), LUE (DCLUE) or WUE (DCWUE). For DCAPAR and DCLUE, tree size was quantified in terms of either stem mass (black points) or leaf area (orange points), while for DCtranspiration and DCWUE, tree size was quantified in terms of either stem mass (black points) or sapwood area (orange points). Data are from Forrester et al. (2012) and Forrester et al. (2013a). Fertiliser application and pruning had no effects on the DC relationships, however thinning effects were evident for DCgrowth - DCLUE or DCWUE (c,d).

asymmetric (DC > 0 or SGR > 1) and belowground competition is size-symmetric (DC = 0 or SGR = 1). However, this can be an inappropriate assumption, as indicated by Schwinning and Weiner (1998) and Casper and Jackson (1997). Asymmetric aboveground competition appears to be based on an assumption that shading by taller trees will cause shorter trees to absorb less light, relative to their size, than taller trees (Weiner, 1990; Schwinning and Weiner, 1998). However, light absorption does not only depend on tree size (e.g. leaf area) and shading, it also depends on crown architecture, leaf angle distributions, leaf morphology and the spatial positioning of taller trees relative to shorter trees (Wang and Jarvis, 1990b; Pearcy et al., 2004; Niinemets et al., 2005; Dong et al., 2016; Forrester et al., 2018). The assumption of size-asymmetric aboveground competition for light may not be appropriate when these vary with tree size or in response to shading, which they sometimes do (Niinemets, 1996; Niinemets, 1998; Niinemets, 2010b; Dong et al., 2016; Forrester et al., 2018). Similarly, to prevent sun damage to foliage, the APAR of larger trees may be reduced, under a high light intensity, by orienting leaves at steep angles and parallel to the direction of the sun (James and Bell, 2000). Therefore, in contrast to the assumption of size-asymmetric above ground competition, there are empirical examples of size-symmetric aboveground competition for light (Binkley et al., 2013; Dong et al., 2016). For example, regardless of which variables are used to quantify size, size-symmetric competition for light (DCAPAR ≤ 0) or size-asymmetric competition for water (DCtranspiration ≥ 0) are shown for several central European species and Eucalyptus plantations in Figs. 4 & 5. In relation to soil resources, competition is sometimes assumed to be size-symmetric (DC < 0 or SGR < 1), but this would require that soil resources are not pre-emptable compared with light and hence that uptake rates are proportional to tree size (Weiner, 1990; Schwinning

and Weiner, 1998). However, belowground competition is not necessarily size-symmetric, because soil water and nutrient availability can be very patchy spatially and temporally (Lehmann, 2003; Schume et al., 2004; Staelens et al., 2004; Hodge, 2006; Gómez-Aparicio and Canham, 2008; Boyden et al., 2012; Uriarte et al., 2015; Christina et al., 2017) and the spatial and temporal distribution of water and nutrient uptake can vary between tree sizes (Weiner, 1990; Schwinning and Weiner, 1998; Christina et al., 2011; Pinheiro et al., in press). Here it is important to consider the spatial and temporal distribution of soil resource uptake, as opposed to root abundance, which is not necessarily correlated with resource uptake (Lehmann, 2003). In situations where plants were considered to be competing strongly for soil resources, sizeasymmetric growth relationships (Rajaniemi, 2003) and uptake relationships (i.e. exponential size-transpiration relationships) have been measured (Forrester, 2015). Fig. 5 shows that competition for water in terms of transpiration, was symmetrical or asymmetrical in Eucalyptus nitens plantations, depending on whether the stands were thinned. Therefore, assuming that aboveground competition is usually sizeasymmetric and that belowground competition is usually size-symmetric is not consistent with empirical evidence. A third and related misconception is that size-growth relationships are reliable indicators of the symmetry or asymmetry of competition for resources, i.e. that DCgrowth is highly correlated with DCuptake. However, while size-growth relationships are partly the result of size-uptake relationships, the production ecology equation indicates that size-growth relationships also depend on size-RUE relationships (Schwinning and Weiner, 1998; Fernández-Tschieder and Binkley, 2018), which can be affected by many processes (Table 1). Size-growth relationships may only be reliable proxies for size-uptake and competition when RUE does not vary with size. Therefore, Fernández-Tschieder and Binkley (2018) 147

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posed the question of whether certain size-uptake or size-RUE relationships are commonly associated with certain size-growth shapes (or DC). In central European forests, while correlations between DCgrowth and DCuptake can be high for some species (e.g. Abies alba in Fig. 4), they were poorly correlated for other species, and in some cases were not correlated (Figs. 4 and 5). Even when DCgrowth and DCuptake were highly correlated, they were not 1:1 relationships, indicating that predictions of DCuptake using DCgrowth will be biased. Even in the absence of competition (open grown plants), DCgrowth could change because uptake and RUE could change in response to age, climate and soil resources. This shows that DCgrowth often cannot be used as a proxy for the symmetry/asymmetry of competition. The question of whether certain size-uptake or size-RUE relationships are commonly associated with certain size-growth shapes has received very little attention (Fernández-Tschieder and Binkley, 2018). While many studies have examined or compared size-growth, size-uptake and size-RUE relationships (Fernández and Gyenge, 2009; Binkley et al., 2010; Forrester et al., 2012; Kunert et al., 2012; Binkley et al., 2013; Campoe et al., 2013a; Campoe et al., 2013b; Forrester, 2013; Forrester et al., 2013a; Forrester and Albrecht, 2014; Gyenge and Fernández, 2014; Forrester, 2015), no common associations have been identified. Theoretically, DCgrowth could be influenced in many ways, including: -

example, a certain structure may be very efficient at obtaining or using water, which will be very useful on sites where there is a low water availability or that experiences a drought, but less useful on sites where there is plenty of water. Few studies have compared given pairs of structures along spatial and temporal gradients, while also accounting for confounding factors like stand density and age. One such example comes from Eucalyptus plantations in Brazil, where size inequality was modified by planting neighbouring trees several months apart (Stape et al., 2010). The planting spacing, genetics and site conditions were all controlled. By the end of the rotation, the more heterogeneous treatment had 14% less wood biomass because stand APAR and LUE were lower (Binkley et al., 2010; Ryan et al., 2010). A similar pattern was observed for several central European species (Bourdier et al., 2016), although this was based on inventory data at many sites rather than paired plots where only the structure differed. The negative effect of size inequality in the Eucalyptus plantations increased with increasing stand density (Soares et al., 2016). This pattern would be consistent with the framework of Fig. 6 if competition for light becomes more intense as stand density (or site quality) increases and the inequality is associated with a reduction in stand APAR and LUE. These were very fast growing Eucalyptus plantations, where a whole rotation could be examined within seven years. Unfortunately, it is much more difficult to examine slower growing species for whole rotations without confounding the effects of disturbances, climatic fluctuations, site, stand density and changes in management or species composition (O'Hara and Nagel, 2006; Pretzsch et al., 2019). Similarly, the structures or management considered interesting a century ago when the stand was established, might not suit modern objectives. An alternative approach for slower growing stands, could be to create different treatments by thinning, thereby ensuring that they all start from the same point. However, a disadvantage is that the tree allometry and physiology would also reflect the past conditions and could influence tree interactions (Table 1), thereby potentially reducing the applicability of the results. When paired plots of contrasting structures are not available along spatial or temporal gradients, an alternative is to combine data from many plots of a given treatment to examine the spatial and temporal dynamics of the variables in Eq. (1), or comparable variables, such as size distribution skewness or DCgrowth. This can then be used to calculate stand growth for each structure along the gradient. This approach was used to examine how growth partitioning (DCgrowth) influences the growth of Picea abies, and whether this changes along a temperature gradient in Switzerland (Fig. 7a). Stand growth increased with DCgrowth but whether stand growth increased along the temperature gradient, also depended on DCgrowth. This shows that responses to climate may not only depend on stand density or species composition or site, but also on growth partitioning, and hence how management influences growth partitioning. More specifically, the same data set was used to compare even-aged forests that often have bell-shaped size distributions, with single-tree selection forests, which have reverse-J shaped size distributions (Fig. 7b). Many studies have compared these stand structures, and a general conclusion is that the productivity is similar, depending on the age of the even-aged stands, or that even-aged stands can be slightly more productive (Kenk 1995; O'Hara and Nagel, 2006). Fig. 7b indicates that comparisons of single-tree selection stands and even-aged stands are strongly dependent on the mean monthly maximum temperature, species mixing (usually with Abies alba) and the age of the even-aged stands. Picea abies has been found to be more productive in mixtures in other central European forests, where mixing effects depend on stand density and site characteristics, but comparisons with Fig. 7b may not be reliable because those studies did not focus on comparing different stand structures (Vallet and Pérot, 2011; Forrester et al., 2013c; Huber et al., 2014). Consistent with the effect of DCgrowth in Fig. 7a, the main cause of the structural effects in Fig. 7b were

Changes in resource uptake or changes in RUE Changes in the response of large trees, small trees, or all trees Changes in the form of increases or decreases Any combination of the above

Any of these appear to be possible, based on the range of tree interactions that can occur (Table 1). This indirectly includes recruitment, mortality and thinning, which change partitioning by adding or removing individuals and by changing the growth of remaining individuals. Therefore, calculations of competition in the form of sizeuptake or DCuptake and the framework of the production ecology equation (Fernández-Tschieder and Binkley, 2018) are valuable tools for understanding the processes of growth or resource partitioning and stand productivity. However, competition asymmetry/symmetry or the resources for which trees are competing (aboveground vs. belowground) cannot be reliably inferred from size-growth relationships. This is because DCgrowth results from many different situations in terms of uptake and RUE (Figs. 4 and 5). Studies that infer competitive relationships from size-growth relationships need to be viewed with caution (Fernández-Tschieder and Binkley, 2018). 4. Temporal and spatial dynamics of stand structure-growth relationships The difference in growth between a given pair of stand structures is likely to vary spatially and temporally with changes in resource availability and climatic conditions. This is because each variable in Eq. (1) can change spatially and temporally, including size distributions (Mohler et al., 1978; Knox et al., 1989; Nord-Larsen and Cao, 2006; Russell et al., 2012; Forrester et al., 2013b; McGown et al., 2016), sizegrowth relationships and DCgrowth (Section 3.3) and stand density. Therefore different structures may be more likely to occur on, or better suited to, different sites and ages. Spatial and temporal changes in how mixed-species forests grow compared with monocultures can be described using a relatively simple framework that may also be applicable for comparing different stand structures (Fig. 6). At the stand level, one structure A could be expected to be more productive than another structure B if it has a higher resource uptake and/or uses resources more efficiently, than B. The framework of Fig. 6 suggests that the relative growth difference between structure A and B could increase with a declining availability of the resource that was improved by that stand structure (Fig. 6). For 148

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Fig. 6. A framework illustrating how the productivity of a given stand structure A relative to another structure B could change along temporal or spatial gradients in resource availability or climatic conditions. The solid red line shows a general pattern where the relative productivity of stand structure A (compared with B) increases as the availability of resource “X” declines (or climatic condition “X” becomes harsher) as a result of a structure that improves the availability, uptake, or use efficiency of resource X (or interactions improve climatic condition X). For example, a structure A that improves water availability, uptake or use could result in an increase in growth compared with structure B, and the relative difference could increase as the availability of water declines. The blue dashed line is a case where the differences in structure do not lead to any change in growth along the gradient because the structure does not influence resource availability, uptake or use efficiency. Modified from Forrester and Bauhus (2016).

differences in size-growth relationships because for the given structures, the shape of the size distributions changed little with time or temperature (Appendix C). Structural equation models have also been used to separate the effects of stand density from species diversity or size diversity (e.g. Barrufol et al., 2013; Vilà et al., 2013). However, interactions with age or site variables need to be included in order to examine the spatial and temporal effects. 5. Is stand growth affected by size inequality or spatial clustering? A lot of attention has been given to the effects of species diversity or species mixing on growth, and the tree interactions that cause such effects (Kelty, 1992; Richards et al., 2010; Forrester and Bauhus, 2016; Liang et al., 2016). However, many of these tree interactions can also be associated with differences in tree size and not necessarily differences between species (Table 1; Forrester and Bauhus, 2016). Given that species mixing effects average an increase in growth of 15–24% compared with the mean of their respective monocultures (Zhang et al., 2012; Jactel et al., 2018), and that contrasting sizes, rather than contrasting species, may contribute part of this “mixing” response (Liang et al., 2007; Lei et al., 2009; Ruiz-Benito et al., 2014; Forrester and Bauhus, 2016), several studies have examined relationships between productivity and size inequality. In such studies, size inequality may be called size heterogeneity, size variability, size hierarchy and size diversity and these are often quantified using the Gini coefficient (Weiner and Solbrig, 1984), Shannon index (Shannon, 1948) or coefficient of variation of tree diameters, heights or other size variables. However, while mixing effects often range between neutral to positive, size inequality effects are often negative (Liang et al., 2007; Stape et al., 2010; Aspinwall et al., 2011; Bourdier et al., 2016; Soares et al., 2016; Yáñez et al., 2017) and sometimes positive (Lei et al., 2009; Ruiz-Benito et al., 2014; Dănescu et al., 2016; Silva Pedro et al., 2017; Ercanli, 2018; Thom and Keeton, 2019). Size inequality is different to all the variables mentioned above because none of the interactions in Table 1 depend on size inequality in the sense that even though some processes require differences in tree size, this can mean that all trees are one of only two sizes as opposed to a diverse range of sizes. Similarly, species mixing effects often occur with only two tree species, and species richness effects often plateau

Fig. 7. The effect of stand structure on growth along a temperature gradient for Picea abies forests in Switzerland (> 400 plots), predicted using linear mixed models and the framework of Eq. (1). As the mean monthly maximum temperature increases, stand growth declines for mean stand structures or stands with low DCgrowth but not for stands with a higher DCgrowth (a). In (a), the stand density and skew of size distributions are held constant at the means in the data set. In (b), the effect of temperature on stand growth is further influenced by mixing with Abies alba or Fagus sylvatica and by management, which is quantified as stands with even-aged bell-shaped size distributions or the reverse-J shaped distributions of selection forests, all predicted for a constant stand density (stem mass of 100 Mg ha−1). Statistical information is provided in Appendix C. To facilitate comparisons in (b), the growth of Picea abies in 1:1 mixtures was multiplied by 2, such that if there was no mixing effect, the mixture and monoculture lines would be the same.

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with as few as five species (Forrester and Bauhus, 2016). Furthermore, for Eq. (1), the shape of a size distribution is important rather than the spread or size range (i.e. the size inequality) of the distribution. Also, the size-growth relationships in Eq. (1) do not depend on the existence of a large range in sizes. There only needs to be a defined size-growth relationship, and that can even be a non-significant relationship (Fig. 1; slope = 0). While these contrasts can appear subtle, they are important distinctions for understanding the causality of inequality effects on growth and other processes. If high size inequality per se could cause large negative effects on productivity, we might expect to find forests with very low size inequality. However, even clonal Eucalyptus plantations can have a coefficient of variation in stem mass of 33% (Luu et al., 2013) and size inequality in these types of plantations is often associated with 10–20% reductions in stand growth (Stape et al., 2010; Soares et al., 2016). Therefore, an alternative hypothesis could be that correlations between stand growth and size inequality exist because size inequality is correlated with other factors that influence stand growth. This might explain the contrasting positive and negative correlations between size inequality and stand growth. Size inequality can be correlated with all of the variables in Eq. (1). For example, with respect to size-growth relationships, size inequality often increases with age under size-asymmetric growth conditions, in the absence of self-thinning, and may decrease with age under sizesymmetric growth conditions (or when self-thinning or thinning occurs) (Weiner and Thomas, 1986; Weiner, 1990; Stoll et al., 1994; Weiner et al., 2001; Dolezal et al., 2004; Nord-Larsen et al., 2006; Dolezal et al., 2009; Bradford and Kastendick, 2010; Castagneri et al., 2012; McGown et al., 2016; Soares et al., 2016; Looney et al., 2018; Sun et al., 2018). Size inequality also increases with increasing stand density (Brand and Magnussen, 1988; Knox et al., 1989; Soares et al., 2016; Sun et al., 2018) and changes with soil resource availability (Wichmann, 2001; Boucher et al., 2006; Metsaranta and Lieffers, 2008; Pothier, 2017; Sun et al., 2018). As discussed in Section 3.3, age and site can both strongly influence size-growth relationships and stand growth. Therefore, size inequality could be positively or negatively correlated with growth because it is correlated with other variables in Eq. (1), such as the sizegrowth relationships. Furthermore, the expected change in size inequality could be determined by considering the predominant types of interactions and which tree sizes are likely to benefit from those interactions (Table 1); if large trees benefit, inequality could increase, whereas if smaller trees benefit, inequality could decrease (in the absence of mortality, thinning and recruitment). Given that so many interactions can influence inequality, its usefulness for predicting stand growth may be limited unless there is a particular process that is dominant in a given forest type such that size inequality is correlated with productivity for a known reason. Size inequality has also been found to correlate with growth after accounting for the effects of density, age and site. For example, in Eucalyptus plantations, stand growth was negatively correlated with size inequality for a given age and density (Luu et al., 2013; Soares et al., 2016). However, the inequality effect depended on tree size, indicating that the inequality effect could be related to the relative height of a tree compared with its neighbours. Relative height or dominance class would be a more direct determinant of growth than the inequality per se. For example, an increase in size inequality of individual tree neighbourhoods negatively affected individual eucalypt tree growth, especially the growth of smaller trees (Luu et al., 2013). The reduction in growth of smaller trees in size heterogeneous Eucalyptus plantations was due to a size-disproportionate reduction in individual tree APAR and LUE (Binkley et al., 2010). Similarly, the growth of tall Quercus robur or Q. petraea trees was positively correlated with height diversity (Shannon index) (Vanhellemont et al., 2018). These studies indicate that in neighbourhoods with higher size inequality, short trees could be more suppressed than they would be in homogenous neighbourhoods, while larger trees could be more

dominant than they would be in homogenous neighbourhoods. Whether this leads to an increase or decrease in stand growth, depends on the variables in Eq. (1). Stand growth of the Eucalyptus stands declined with increasing size inequality because the small to intermediate size classes not only grew less but they possibly also contributed a large proportion of the stand density. This is assuming that the size distributions were similar to the bell-shaped distributions in Fig. 1, which is often the case for young unthinned Eucalyptus plantations (Forrester et al., 2013b; Luu et al., 2013; Scolforo et al., 2019). The increased growth of dominant trees, due to their higher APAR and LUE (Binkley et al., 2010), was not enough to overcome the growth reduction of the smaller trees because the large trees contributed a relatively small proportion of the stand density. Besides size heterogeneity, forests can be heterogeneous in terms of the spatial clustering of trees. Spatial clustering can result from environmental gradients within the stand, spatially-dependent regeneration or self-thinning and from management or disturbances that create gaps (Liu and Burkhart, 1994; Stoll et al., 1994; Gersonde et al., 2004; Aakala et al., 2013; Looney et al., 2018; Resende et al., 2018). As trees compete and self-thinning occurs, the spatial positioning of trees often becomes more uniform (West, 1984; Kenkel, 1988; Kenkel et al., 1997; Svoboda et al., 2010; Zeller and Pretzsch, 2019), although disturbances may change this pattern, particularly in older stands (Larson et al., 2015). Spatial clustering may influence individual tree growth by creating high-density neighbourhoods and low-density neighbourhoods within a single stand. As a result of these high-density neighbourhoods, individual tree growth can be reduced by spatial clustering (Stiell, 1982; Adams et al., 2013; Looney et al., 2018), and stand growth is also sometimes reduced (Stiell, 1982; Resende et al., 2018). These growth reductions may be mitigated against by the plasticity in crown positioning (or root distributions). That is, to obtain more favourable canopy positions, individual trees can develop asymmetric crowns, which may no longer be centred above the point where their stem is in contact with the ground (Stiell, 1982; Umeki, 1997; Brisson, 2001; Longuetaud et al., 2013; Lee and García, 2016). Similarly, in mixed-species stands, different species can be clustered in small monospecific groups or mixed more intimately on a tree-by-tree basis. If intra-specific competition is more intense than inter-specific competition, then a less clustered, more intimate mixture can be more productive (e.g. Bieng et al., 2013), but this is not always found (Forrester and Smith, 2012). Therefore, the clustering of species is often used to manipulate the interactions between the species in plantations (Kelty, 2006). Few studies have examined how spatial clustering influences the variables in Eq. (1). Looney et al. (2018) found that spatial clustering reduced the growth of larger individuals. Additional studies that examine which interactions in Table 1 are modified by spatial clustering, may help to elucidate how spatial clustering influences tree and stand growth. It is conceivable that spatial clustering effects on variables such as size-growth relationships may be in the form of an increase in the residual variability within the size-growth relationship, rather than on the relationship itself. Therefore, an alternative approach could be to calculate neighbourhood indices for all individual trees (e.g. sum of tree basal area within 10 m) and divide the trees into groups based on their neighbourhood indices. The variables of Eq. (1) could then be examined for each of these groups (high or low density neighbourhoods) to identify the dynamics within the clusters compared with the gaps. The structural variables in Eq. (1) as well as site, climate, age and dominance class are important determinants of tree growth. However, based on the studies in this section, size inequality may less directly influence growth, and instead be correlated with growth because it is correlated with other factors that influence growth. Therefore, size inequality may need to be used with caution when trying to understand how stand structure influences forest growth. 150

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6. Implications for quantifying stand structure

is used, as in Eq. (1), the contrasting structure-growth patterns from different studies appear to be more consistent. This cannot be definitively tested in this review because most studies do not provide all variables in Eq. (1), especially the size distributions. Nevertheless, each structural variable in Eq. (1) is clearly important for quantifying different structural characteristics that influence growth. The size distributions quantify the vertical structure and how the stand density is distributed. The size-growth relationships and DCgrowth describe the partitioning of growth and the likely future development of size distributions (Fig. 9). The size-uptake and size-RUE relationships explain the size-growth relationships. Measurements of the processes in Table 1, in order to calculate DCuptake or DCRUE, or relative height, may be very helpful in identifying the processes behind the patterns (e.g. Figs. 4 and 5). On the other hand, the use of too many variables can also be problematic. Instead, Eq. (1) could be used as a framework to focus on a smaller number of variables that can have clear effects on growth.

In this review, an attempt was made to describe stand structure using variables that are easy to measure, calculate and interpret and that can potentially quantify responses to factors such as climate, species mixing, resource availability, density and management. While some stand structural terms can be ambiguous e.g. “canopy stratification” (Parker and Brown, 2000), many indices have been developed to directly quantify structure (Staudhammer and LeMay, 2001; Williams and McCarthy, 2001; Pommerening, 2002; von Gadow et al., 2012; del Río et al., 2016). Before using any index to quantify stand structure or as a proxy for other processes, it is critical to consider what they actually indicate (McElhinny et al., 2005; Köhl and Baldauf, 2012). This is important because a given index can represent multiple processes in Table 1, some of which have opposing effects on growth or are correlated with growth in different directions depending on the circumstances e.g. DC and size inequality can be positively or negatively correlated with growth (Sections 3.3 and 5 and Fig. 3). Generally, the effects of stand structure on growth described in this study could not be explained by only using a single structural variable (e.g. stand density, size inequality, DC, SGR, size distribution shape) as also indicated in Fig. 8. Similarly, Ishii et al. (2004) concluded that instead of aiming to develop a single index of stand structural complexity, which may be only weakly correlated with the process of interest, it is more useful to use one that quantifies the processes causing the effects. Based on the studies reviewed above, when > 1 structural variable

7. Implications for sampling and scaling from trees to stands Scaling up from a sample of trees to the stand generally requires that the variables in Eq. (1) are accounted for. This is often done automatically by randomly selecting a sample of trees that includes all size classes, species etc. This accounts for all variables in Eq. (1) without actually quantifying any of them individually. While this is a common approach in many areas of forest ecology, it is not always the case and several dendrochronology reviews have shown how different sampling approaches can lead to large and variable biases compared with when all trees are measured (Cherubini et al., 1998; Carrer, 2011; Nehrbass-

Fig. 8. Effect sizes of independent variables in Eq. (1) on stand stem mass growth. The y-axis indicates how many standard deviations the stand growth will change if the variables on the x-axes increase by 1 standard deviation (σ). The effect sizes are quantified as the slopes of the variables on the x-axes, all of which have been standardised (mean = 0 and σ = 1) and fitted to linear mixed models for eight species in central Europe (the statistics are described in Appendix D). Each variable is shown on the x-axis in terms of its mean effect and how it interacted with a second variable that is indicated in the parentheses. The second variable (in parentheses) is decreased or increased by 1 σ. When the interactions were not significant, only the mean bar is shown. The skewness of size distributions often had a minor effect because the majority of the stands had bell-shaped size distributions (with the exceptions of Abies alba, Picea abies and Fagus sylvatica, which also occur in single-tree selection forests). 151

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factor (Fig. 7a). A random sample, or sampling all trees, is also necessary for calculating the DC or SGR, which both require a sample of the size range in proportion to the abundance of each size; it is not enough to just sample the range of sizes. With regards to understanding the responses, Fig. 1 shows that even for a given density, stand growth can differ greatly (> 100%) only due to differences in size distributions or differences in individual tree sizeresponse relationships. Therefore, inferences about the stand mean (or sum) based on samples that only include a subset of sizes (e.g. largest trees) will be unreliable unless all sizes responded in the same way to the given factor. Similarly, inferences about the stand mean (or sum) based on a sample covering the size range, but not a random sample (in proportion to the abundance of each size class), will lead to biased results unless the distribution of the given variable is symmetrical (e.g. perfectly bell-shaped). 8. Management implications Silvicultural practices are often based on manipulating or maintaining certain size distributions, the partitioning of growth or resources between different size classes and the stand density. Therefore, the variables in Eq. (1) are often foundational for quantifying silvicultural treatments (O'Hara, 2014) and provide a framework for predicting the effects of, and to refine, silvicultural treatments. For example, the effects of climate, species mixing and age on comparisons between even-aged and single-tree selection forests were examined in Fig. 7b. That is, Eq. (1) can be used as a framework to quantify how management influences stand structure and therefore how and why different management can influence responses to climate (Fig. 7). While there are many potential silvicultural objectives relating to thinning, the following examples relate to manipulating how growth is partitioned between trees. The partitioning of resources between trees is often considered when thinning, and therefore, the DCgrowth has been recommended as a tool to determine which trees to thin and whether previous thinning interventions achieved their goals, in terms of the redistribution of growth or resources within the stand (Bradford et al., 2010; Doi et al., 2010; Trouvé et al., 2014; Pothier, 2017, 2019). Thinning from below In even-aged Eucalyptus and Pinus plantations with DCgrowth > 0, thinning from below could be used to focus growth on the largest and most efficient trees, which are often the most valuable in such plantations, and this may be particularly important where resources are limited and a higher RUE is desired (Binkley et al., 2010; Doi et al., 2010; Campoe et al., 2013a; Forrester, 2013). However, the size distributions also need to be considered, because the shape of diameter distributions can be as important as factors such as site quality in determining thinning responses (Forrester et al., 2013b); thinning responses of dominant trees in even-aged Eucalyptus plantations can be higher when size distributions contain high proportions of large trees (left skewed) and lower for right skewed distributions, where dominant trees are less likely to be competing with other large trees.

Fig. 9. An example of how the black bell-shaped size distribution will change into the red size distribution as a result of different size-growth relationship shapes, similar to those in Fig. 1. Modified from Westoby (1982).

Ahles et al., 2014). Biased results come from sampling strategies that only sample the range of sizes as opposed to random sampling, which samples the range of sizes in proportion to the abundance of each size class. The greatest errors result from only sampling a part of the size range and then using means or sums to make inferences about the total stand (Nehrbass-Ahles et al., 2014). While not explicitly quantified in these reviews, the degree of bias will depend on the variables in Eq. (1), especially the size distributions, as illustrated in Fig. 1. The framework of Eq. (1) can therefore be used to refine or complement sampling and scaling approaches and also to show why stands that seem very similar (e.g. same basal area and size range) can respond differently to a given

Thinning from above or thinning all sizes In stands where DCgrowth = 0, any size class could be targeted for thinning without decreasing its future growth potential, at least in the short term (Pothier, 2017). In contrast, when DCgrowth < 0, thinning from above could be used to remove some of the large, less efficient, trees while retaining smaller trees that do not compete with retained large crop trees in order to maintain large-tree growth while simultaneously promoting the growth of the residual stand (Looney et al., 2018). 152

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Assessing the effect of previous thinning

reduced. A classic example where this is already practiced is when a shade tolerant species is mixed with a less tolerant species. The taller individuals of the shade tolerant species are removed so that they do not suppress the less shade tolerant species. Nevertheless, species interactions are not always so obvious and the framework of Eq. (1) can help to manage such interactions.

The DCgrowth has also been used to indicate how effective thinning has been (Bradford et al., 2010; Soares et al., 2017), as opposed to how effective thinning might be. In these studies, thinning from below or from above and below was found to reduce the DCgrowth in Pinus resinosa and Eucalyptus stands, often to nearly 0. This indicates that the thinning interventions were successful at retaining trees that made similar contributions to stand growth relative to their size. The thinning presumably removed the less efficient trees thereby enabling the retained trees to maintain or improve their efficiency, in terms of growth relative to size.

9. Conclusions Many effects of stand structure on growth may be described using a simple framework that considers stand density, size distributions and size-growth relationships. Although these variables can be positively or negatively correlated with growth, when all considered together, there could be a greater convergence of the apparent contrasting growthstructure correlations. In contrast, size inequality does not appear to be as useful for quantifying the effects of stand structure on growth. In addition to growth, Eq. (1) could be used to study other forest processes where the process depends on tree size (e.g. resource use, drought stress, species mixing). Similarly, Eq. (1) can be used to split the effects of structure from functioning when examining silvicultural treatments, spatial and temporal gradients in resources and climate and when comparing mixed with monospecific forests. Future work could test whether the framework of Fig. 6 is appropriate for describing the spatial and temporal dynamics of how stand structure influences growth.

Predicting future size distributions from current growth partitioning As implied in the term “distribution-modifying function”, the shapes of the size-growth relationships determine how the size distributions will change (Westoby, 1982). While the framework of Eq. (1) indicates how the current structure is influencing growth, the potential future size distributions (in the absence of thinning or mortality) can be predicted from the size-growth relationships (Fig. 9). It is important to note that as the distributions change, so will the size-growth relationships and the stand density, so Fig. 9 is only indicative. Predicting stand growth from growth partitioning

Acknowledgements

As shown in Section 3.3 (Fig. 3), DCgrowth can be positively correlated with stand growth in some stands but negatively correlated with stand growth in others. Therefore, this correlation is not useful for predicting stand growth unless there is already knowledge of the direction of the relationship for the given population, age and growing conditions. Positive correlations between stand growth and DCgrowth will result when partitioning changes in favour of the tree sizes that are contributing a higher proportion of the stand density (Fig. 1).

The data used to make Figs. 3, 4, 7 and 8 were from the Swiss longterm plot network (Experimental Forest Management project) and therefore the long-term commitment of WSL and many scientists and technicians who collect these data is gratefully acknowledged, in particular, Jens Nitzsche and Hubert Schmid. Alvaro Soares, Andreas Brunner, Alessandra Bottero, Paolo Cherubini and an anonymous reviewer provided comments that improved upon an earlier version of the manuscript.

8.1. Mixed-species forests

Appendix A. Supplementary material

The effects of species mixing on forest growth can result from changes in processes but also from changes in structure (Forrester and Bauhus, 2016; Madrigal-González et al., 2016; Pretzsch et al., 2016; Riofrío et al., 2017). Different species vary in their response to different structural characteristics (Fig. 8). For example, while most central European species in Fig. 8 were found to be influenced by stand density, some were similarly effected by growth partitioning (DCgrowth) whereas others were not. These contrasting responses to structure could be modified or exacerbated by species interactions. The framework of Eq. (1) can facilitate the identification of the structural attributes that contribute most to mixing effects and therefore how to manage the stand structure in relation to species composition. All components of Eq. (1) have been shown to change in mixtures compared with monocultures and species interactions may strengthen many of the processes in Table 1. That is, size-growth, size-uptake or size-RUE relationships can change in mixtures (Kunert et al., 2012; Forrester and Albrecht, 2014; Forrester, 2015). Similarly DCgrowth can differ between mixtures and monocultures (Binkley et al., 2003b; Pretzsch et al., 2016). Stand density can change due to mixing effects on growth and mortality (Binkley, 1984; Woodall et al., 2005; Weiskittel et al., 2009; ReyesHernandez et al., 2013; Pretzsch and Zenner, 2017). The shapes of size distributions can also be influenced by mixing (Binkley, 2003; Binkley et al., 2003b; Forrester et al., 2004; Forrester and Smith, 2012), and in some cases appear to have been the main contributor to mixing effects (Forrester and Smith, 2012). This information could then be used to modify or develop silvicultural regimes that target particular structural characteristics of particular species on certain sites. Favoured species interactions could be targeted while the effects of competitive interactions could be

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