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Modeling internode length and branch characteristics for Pinus radiata in New Zealand
Forest Ecology and Management 160 (2002) 243–261
Modeling internode length and branch characteristics for Pinus radiata in New Zealand R.C. Woollonsa,*, A. Haywoodb, D.C. McNicklec a
School of Forestry, University of Canterbury, Christchurch, Canterbury, New Zealand b Carter Holt Harvey Forests, P.O. Box 648, Tokoroa, New Zealand c Department of Management, University of Canterbury, Canterbury, New Zealand Received 8 October 2000; accepted 2 February 2001
Abstract It is recognized that estimation of internode length and maximum branch size is important for the prediction of clearwood in unpruned timber stands, as well as for evaluating the quality and value of logs in general. A review of existing branch models reveals a diversity of approaches as well as a tendency for results to be species specific. Here, a branch model is developed for Pinus radiata in New Zealand, capable of predicting successive internode lengths, the number of branches on each branch cluster, and the size of each branch up to the green crown (GC) at site index age 20. Inputs to the model include tree height and diameter at breast height (dbh) (both at age 20) and basal area per hectare of the top 100 stems. Further optional inputs are an ocular count of the number of branch clusters up to the green crown, and branch factor (BF)—the size of the biggest branch in the first cluster encountered above 6 m. The vertical distributions of internode length and maximum branch diameter are found to reach maxima around 0.3–0.4 of relative height. Internode length and the number of branches per whorl were found to be independent of tree size, site index and stand density. Stems per hectare is not required as an explicit predictor variable but it appears implicitly through tree dbh. Some model output is given and the results are discussed. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Radiata pine; Branch dimensions; Distance between branch clusters; Predictive simulator
1. Introduction The study of tree branches is certainly not new. Leonardo da Vinci1 (c1500), hypothesized that the sum of branch cross-sectional areas above any branch cluster should be equivalent to the cross-sectional area of the bole below that cluster. For Pinus radiata in Australia, Jacobs, 1938 presented a monograph on branch habits that influenced the efficiency of pruning. * Corresponding author. E-mail address: [email protected] (R.C. Woollons). 1 see MacCurdy (1954).
Fielding, 1960 gave a description of branch morphology for radiata pine around Canberra in Australia, while Bannister, 1962 reported the species in New Zealand, concentrating on annual shoot and branch development from a physiological viewpoint. More recently, several authors have stressed the importance of predicting tree internode length2 and how it may 2 Bannister (1962) points out foresters have developed ambiguous definitions of the terms ‘node’ and ‘internode length’ that can be at variance with physiological and botanical viewpoints. From a modeling perspective, the differences in definition are unimportant and we continue to use the terms. This should not be taken as any rebuttal of Bannister’s stance.
0378-1127/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 1 1 2 7 ( 0 1 ) 0 0 4 6 8 - 6
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vary with breed, site, and management, because it is an important variable in determining amounts of clearwood which can be obtained from unpruned logs (Carson and Inglis, 1988; Grace and Carson, 1993). Many current end-uses for P. radiata clearwood require lengths below 2 m and often less than 0.5 m. At the same time, branch size and number are also well recognized to be vital factors in assessing quality and value of timber (Brazier, 1977; Whiteside, 1982). These points lead to the feasibility of predicting internode length, cluster number, and the sizes of branches thereon by some simulation system. Apart from being a potential asset in evaluating various silvicultural regimes it is seen as a critical tool to enhance the utility of bucking algorithms by providing more information to assess log value (Pnevmaticos and Mann, 1972; Pnevmaticos, 1975; Greets, 1984; Eng et al., 1986). There are numerous branch models described in the literature (see, among many, Cochrane and Ford, 1978; Maguire et al., 1994; Makinen and Colin, 1998) but comparisons between them quickly become complex and diffuse. The response variables, branch angle, branch size, whorl numbers per annual shoot, vary from one study to another as can the sampling units chosen-annual shoot, whole tree, crown, primary or secondary branch. Moreover, results are strongly dependent on species. Maguire et al., 1994 comment ‘. . . model characteristics vary as a result of differences among species. . . specifically in the relative prominence of nodal verses internodal branches’. Whereas P. radiata and Pinus sylvestris, for example form one or more distinct nodes per year with no internodal branches, Chamaecyparis produces branches regularly along the stem and Pseudotsuga is intermediate between these patterns. It follows that useful comparisons between models should be limited to the same species and to models with similar objectives. Two branch models exist for radiata pine in New Zealand. Inglis and Cleland, 1982 built a multiple regression model which predicts branch index defined as the mean of the largest four branches for a nominated log length, taking one from each of four cross-sectional quadrants (regarding the log as a cylinder), at nominated log height classes. Log lengths are required to be between 5 and 6 m. The model uses four stand-level predictor variables; site index (mean-top-height at age
20), mean-stand-diameter at age 20, height class (of nominated logs), and mean-top-height at any (previous) thinning. The model error is calculated as 37 mm with a R2-value of 0.92, although this figure is inflated by the pooling of several thousand data. Grace et al., 1998 discuss the structure of a branch model where the numbers of branch clusters are predicted in relation to annual shoot growth. No actual formulae are given but emphasis is placed on mechanistic principles, and total branch composition and internodal structure are obtained by summing the results of the annual shoot development after growing the branches to maximum size. We reviewed these models and decided that neither suited our objectives. The multiple regression model is somewhat dated and gives virtually no information on either internode length or individual branch size, while the process-based approach assumes that annual shoot length can be satisfactorily and precisely modeled an assumption that seems doubtful considering the complexity of shoot development (Bannister, 1962; Burdon, 1994). We were also influenced by measurement considerations. Adoption of a mechanistic model meant that the data would require the collection of stem analysis material, a complex and time consuming exercise (McEwen, 1975) compounded by a likelihood that shoot discontinuities would be difficult to assess with any certainty (Burdon, 1994). Accordingly it was decided to ignore shoot and branch growth. Instead, for a given tree, we would estimate the number of dead branches comprising any cluster, the respective (final) branch sizes, and the distance between any two consecutive clusters (internode length) up to the green crown (GC). In this contribution we describe the construction of such a P. radiata branch model for New Zealand North Island plantations. The construction and rationale of the component equations, together with the model logic are described and some output from the system is presented. Some limitations and future developments of the model are discussed.
2. Data We wanted the model to include both stand and tree input measures. These required the data to be collected in plots and at a fixed (index) age because although
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lower-stem branches have ceased to grow (maximum size is achieved in around 10 years for P. radiata, Brown, 1962; Grace et al., 1998), stem growth continues to develop. Age 20 was chosen as the index— the age New Zealand foresters use to define site index for P. radiata (Goulding, 1986). Moreover, it represented a logical age to carry out stand inventories being only 4–6 years before clearfall. Data were collected from 191 trees in three Carter Holt Harvey Ltd. (CHH) forests (Mahurangi, Riverhead, and Kinleith) in the North Island of New Zealand. At Mahurangi, a high production forest, two 20-yearold permanent sample plots were clearfelled and 60 trees sampled. At Riverhead, a lesser productive area (Ballard, 1977) a single permanent plot of 43 trees was sampled. At Kinleith a more elaborate sampling strategy was employed. The forest was stratified into four large compartments and in each, 20-year-old stands were selected to secure a range of stockings. Six to eight 1/10 ha plots were established in each selected stand and within each, three trees were extensively measured. Residual trees were only measured for diameter at breast height (dbh) outside bark. No thinning had been carried out in any plot later than age 8.
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All trees measured for branch characteristics were sampled as follows. Before felling, an ocular estimate of the number of branch clusters to the GC was obtained, along with a measure of branch factor (BF), defined as the largest branch in the first whorl occurring above 2 m. After felling and measuring tree dbh and total height, the distances between all branch clusters from the butt to the tip were obtained (at Kinleith, they were measured only to the GC). The distance to the GC height was recorded, defined to be the distance to the first branch cluster containing at least one branch with green needles. For all branch clusters, the maximum branch diameter was measured with calipers, directly adjacent to the stem. For every fifth cluster all branches were measured for diameter, regardless of size. Table 1 summarizes these data.
3. Component models 3.1. Maximum branch size 3.1.1. Changes in vertical distribution A plot of average maximum branch size and relative height class shows a pronounced ‘bulge’ at around
Table 1 Summary of branch, tree and stand data Mean
Standard deviation
Minimum
Maximum
Kinleith Stems per hectare Diameter at breast height (dbh, mm) Tree height (m) Site index (m) Maximum branch (mm)
531 353 25.7 28.7 34
241 136 5.5 3.3 20
260 111 13.2 22.9 2
1020 592 37.0 34.2 118
Mahurangi Stems per hectare Diameter at breast height (dbh, mm) Tree height (m) Site index (m) Maximum branch (mm)
488 388 29.5 32.1 41
10 78 3.1 1.6 18
480 210 20.7 30.1 3
500 529 37.2 33.5 115
Riverhead Stems per hectare Diameter at breast height (dbh, mm) Tree height (m) Site index (m) Maximum branch (mm)
430 329 27.3 29.6 26
0 60 3.3 0 13
430 129 18.5 29.6 1
430 475 32.5 29.6 95
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Fig. 1. Relationships between average maximum branch size and relative height of the tree.
0.3–0.4 of relative height then decreasing thereafter. (There are relatively few data above relative height around 0.6 since these are generally above GC height where the branches are still growing.) The relationships for the three forests are depicted in Fig. 1. A function MAXBRrel ¼
hr þd ða þ bhcr Þ
(1)
where MAXBRrel is the relative maximum branch size, defined to be any maximum branch size on the ith tree, as a proportion of the average (maximum) branch size for the ith tree, where the average is calculated over all whorls, hr the relative height along the tree; a, b, c and d are regression coefficients, estimated by non-linear least-squares, was fitted to the data. All coefficients are significantly different from zero, as determined by asymptotic 95% confidence intervals and the lower bounds of these are a minimum of four deviations from zero. The residual values resulting from (1) are unbiased, when compared with predicted values, relative height and forest.
Fig. 2 shows the cumulative ordered residuals plotted against a normal distribution with zero mean; clearly the overall goodness-of-fit is very good. The first attempts to model this effect with variants of Eq. (1) used absolute rather than relative maximum branch size. It was found that this could generate extraordinary small or even negative branch sizes. Usage of relative branch size removed this problem. 3.1.2. Estimation of maximum branch size From Table 1, maximum branch size is strongly (and logically) related to stem diameter. Fig. 3 shows the distribution of branch sizes, scaled by dividing each by the predicted value of (1) given the appropriate relative height for the three forests. Preliminary analyses of the maximum branch distribution showed it was best modeled by the Weibull distribution c x yc1 xy c f ðxÞ ¼ exp ; x>y s s s (2)
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Fig. 2. Goodness-of-fit of residuals for the branch relative height model.
Fig. 3. Frequency distributions of maximum branch size for the three forests.
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where y, s, and c are location, scale, and shape parameters (s > 0, c > 0). Fitting (2) to distribution data can be achieved by a variety of estimation techniques (Bailey and Dell, 1973). Maximum likelihood estimation of such a shifted distribution can be problematic (Law and Kelton, 1991) so we utilized the method of moments but avoided requiring an iterative solution by adopting an explicit approximation proposed by Garcia, 1981. This requires the estimation of the mean, standard deviation, and minimum scaled maximum branch size. After much analysis, six models were adopted, depending on whether the variable BF, defined above, is incorporated as a predictor variable. The figures in brackets below each equation give the significance level for the hypothesis that any coefficient is equivalent to zero. 1. Mean scaled maximum branch size MEANMAX ¼ b1 d20 þ b2 h20 þ b3 BA100 ðall < 0:0001Þ; R2 ¼ 0:96;
s2 ¼ 47:6
(3)
MEANMAX ¼ b1 d20 þ b2 h20 þ b3 BF R2 ¼ 0:98;
s2 ¼ 30:2
3.2.1. Changes in internode length by latitude There is some evidence in New Zealand that internode length may be a function of latitude and site (Carson and Inglis, 1988; Grace and Carson, 1993), with a tendency for the most southerly plantations to have longer internode lengths. Close examination of these studies reveals that the database on which the conclusions are based is very small (one plot in each of 16 forests throughout the country), and thus, some doubt must exist as to the strength of the relationships claimed. Consequently we decided to ignore this facet of internode behavior in the model. Table 2 and Fig. 4 summarize the internode length data for the three
(4)
ð0:01Þ
ð0:025Þ ð0:001Þ
s2 ¼ 20:6
(5)
STDMAX ¼ b1 expðb2 d20 Þ þ b3 BF ð0:005Þ
ð0:025Þ ð0:01Þ
þ b4 MEAN BF; ð0:025Þ
R ¼ 0:45;
(9)
3.2. Estimation of internode length
STDMAX ¼ b1 expðb2 d20 Þ þ b3 BA100
2
MAXBR ¼ ððlogð1 rÞÞ1=c Þs þ MINMAX
þ b4 MEAN BF ðall < 0:0001Þ; 2. Standard deviation of scaled maximum branch size
R2 ¼ 0:37;
where in (3)–(8) above: (MEAN, STD, MIN) MAX are mean, standard deviation, and minimum scaled maximum branch size; BA100 is basal area per hectare of the top 100 trees; MEAN_BF the average (plot) BF; d20 the dbh, at age 20 (mm); h20 the total stem height, at age 20 (m), and all other terms are defined earlier. The residuals of the six models are without exception, without serious bias. The models are devoid of intercepts to avoid aberrant behavior near zero. To estimate maximum branch sizes for a given tree, appropriate inputs in (3)–(8) (dependent on whether BF has been measured), give estimates of the mean, standard deviation, and minimum (maximum) branch size. Using Garcia’s moment procedure allows direct calculation of scale and shape parameters in (2) (the location parameter being estimated by MINMAX through (7) or (8)) thus creating a unique Weibull distribution for each tree. A maximum branch estimate (MAXBR) is immediately available through the inverse cumulative distribution form of the Weibull after generating a random uniform deviate (r)
2
s ¼ 18:1
(6)
3. Minimum scaled maximum branch size Table 2 Summary of internode lengths (m) by forest
MINMAX ¼ b1 d20 þb2 BA100 ð0:0001Þ 2
R ¼ 0:82;
ð0:0133Þ
2
s ¼ 24:4
(7)
MINMAX ¼ b1 d20 þ b2 MEAN BF ð0:0001Þ
R2 ¼ 0:82;
s2 ¼ 24:2
ð0:0067Þ
(8)
Forest
Average
Standard deviation
Minimum
Maximum
Kinleith Mahurangi Riverhead
0.60 0.55 0.50
0.38 0.28 0.30
0.08 0.04 0.04
2.38 2.30 2.34
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Fig. 4. Frequency distributions of internode length for the three forests.
regions. Kinleith is some 300 km south of the other two forests but the difference in average internode length is not statistically significant. 3.2.2. Changes in vertical distribution When the mean-tree internode lengths were plotted against relative height classes (Fig. 5), a clear systematic trend appeared with generally shorter lengths around the butt, but sharply increasing to around 0.30 of tree height, then gradually decreasing thereafter to the tip. (The effect is unclear when individual lengths are graphed against actual relative heights for all trees because of extreme variation.) A function ILrel ¼
hr þd a þ bhcr
(10)
where ILrel is relative internode length defined to be any internode length on the ith tree, as a proportion of the average internode length of the ith tree; hr the relative height along the tree, a, b, c, and d are regression coefficients estimated by non-linear
least-squares, was satisfactorily fitted to these data. All coefficients were significantly different from zero, as assessed by 95% asymptotic confidence intervals and the lower bounds of these are a minimum of three deviations from zero. Residual values resulting from (10) were unbiased, when compared with predicted values, relative height and forest. Fig. 6 shows the cumulative ordered residuals plotted against a normal distribution with zero mean; the overall goodness-offit is satisfactory. As for maximum branch size the use of relative rather than absolute internode length negated the occasional generation of lengths around zero. 3.2.3. Serial correlation of internode lengths We explored possible serial correlation (Draper and Smith, 1981) between successive internode lengths by fitting model (10) individually to each of the 191 trees, then treating each sequence of residual values as a time series. A first order autocorrelation coefficient (Chatfield, 1985) was calculated for each tree and the results compared. Table 3 summarizes these values.
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Fig. 5. Relationships between internode length and relative height of the tree.
From Table 3 it is concluded that no significantly large serial correlations exist between successive internode lengths. Apart from the low average values there is evidence that the higher values for Kinleith in part may be a function of too few internode numbers-these trees being only measured to GC height. 3.2.4. Relations of internode length with growth and space The internode lengths were scaled by dividing by the predicted value of (10) given the appropriate relative height. These were averaged for each tree, and Pearson correlation coefficients were calculated for various tree and stand variables. Table 4 summarizes these statistics. Although some coefficients are significantly different from zero, for practical purposes the correlations are very weak and can be disregarded. To explore any stand effects for internode distance, a nested random effects analysis of variance was applied to the Kinleith forest data. Results are given in Table 5.
The very dominant source of variation is attributable to fluctuations along the stem. A second but appreciably smaller source of variation might be attributable to tree-to-tree differences, probably reflecting distinct genetic composition (Grace and Carson, 1993). 3.2.5. Prediction of internode length From the above analyses it is evident internode length can be regarded as a random phenomenon apart from systematic effects occurring in relation to relative height and perhaps differences occurring with different trees. To simulate internode length on any stem we created two distributions: 1. the distribution of scaled mean-tree internode distances; 2. the distribution of scaled deviations (residuals) around mean-tree values. Both distributions (1) and (2) are asymmetric, being (logically) skewed to the right. Several probability distribution functions were considered to model these distributions beta, gamma, log-normal, and
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251
Fig. 6. Goodness-of-fit of residuals for the internode relative height model.
Weibull, but after comprehensive analyses and applying various goodness-of-fit procedures the lognormal distributions ! 1 ðlogðx yÞ zÞ2 exp f ðxÞ ¼ pffiffiffi ; 2s2 s 2pðx yÞ x>y
(11)
where s, y, and z are estimated shape, scale, and location parameters (s > 0) and (1) and (2) referring Table 3 First order autocorrelation coefficients of residuals for each forest Forest
Average
S.E.
Minimum
Maximum
Kinleith Mahurangi Riverhead
0.240 0.122 0.006
0.035 0.028 0.028
0.27 0.34 0.33
0.56 0.49 0.39
to the mean and deviation distributions, were adopted. Figs. 7 and 8 show the respective distributions and the fitted log-normal probability functions. The fit of the mean-tree internode lengths is very satisfactory with the Anderson–Darling goodness-of-fit statistic being completely non-significant (P < 0:4). The modeling of the deviations of internode lengths is less precise but still reasonable. To produce internode distances for a specific tree the following algorithms are invoked, dependent on whether an ocular estimate of the number of branch clusters to the stem GC is available. 3.2.6. Prediction of internodes with no estimate of whorl numbers available An average tree internode length (AVEL) is generated from the appropriate log-normal distribution (11)
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Table 4 Correlation coefficients for the scaled internode distances Tree diameter at breast height (dbh)
Tree height
Site index
Stems per hectare
Basal area per hectare
0.193
0.125
0.036
0.035
0.151
Table 5 Nested ANOVA for Kinleith internode lengths Source of variation
D.F.
Mean square
Total variation (%)
Sub-forests Plots within sub-forests Trees within plots Internodes within trees
3 26 60 1904
1.904 0.573 0.364 0.127
1.89 1.95 7.75 88.41
above using standard random-variate generation techniques and the log transformation described in Law and Kelton, 1991. Having generated an average tree value successive internode lengths are generated (for a specific tree) by similarly producing internode deviations (Idi) from (2) above. A series of scaled internode lengths (SILi) are then obtained from SILi ¼ AVELi Idi
Fig. 7. Histogram and fitted normal distribution of residuals for the mean-tree branch model.
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253
Fig. 8. Histogram and fitted normal distribution of residuals for the deviations branch model.
This sequence of scaled internode lengths can be converted to actual lengths by multiplying each SILi value by the appropriate relative length value in (10). These are accumulated and the process is repeated until the total internode lengths equal or exceed the (known) height of the tree.
3.2.7. Prediction of internodes with an estimate of ocular count available When an estimate of the number of branch clusters to the GC (NCGC) is available the estimation process above can be improved.
A model of GC height is given by GCh d20 Cratio ¼ ¼ b1 þ b2 exp h20 1000
(12)
where Cratio is the ratio of green crown height (GCh) to total stem height at age 20 (h20); d20 the dbh, at age 20 (mm); b1 the regression coefficients. Model (4) has an R2 ¼ 0:57 with unbiased residuals, both with respect to predicted and d20 values, while both coefficients are significantly different from zero (P < 0:0001). Usage of (12) allows the estimated NCGC to be converted to an estimate of the total number of branch clusters on the stem (NCS) this
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same number of internode lengths are then generated as for (Section 3.2.6) above. In general, the accumulated lengths estimated will not be equal to total tree height. This is achieved by scaling each length by the factor NCS/h20.
create different models for each number. These distributions proved quite difficult to model with several exhibiting small multi-modal characteristics. Ultimately, we fitted a beta density function to the relative diameters to each class.
3.3. Estimation of branch numbers
f ðxÞ ¼
Preliminary analyses showed that branch cluster numbers were independent of tree size, stems per hectare or site index. For our data branch numbers on clusters varied from 1 to 14 with a mode of five. The frequency distribution created was modeled by a truncated Poisson probability distribution (Johnson and Kotz, 1969). Prðx ¼ kÞ ¼ expðlÞð1 expðlÞÞ1 lk =k! k ¼ 1; 2; . . .
(13)
where the parameter l was estimated by an explicit expression given by Irwin, 1959. A Kolmogorov– Smirnov goodness-of-fit test (P < 0:13) substantiated a reasonable fit to the data. An estimate of branch number is obtained by accumulation of the respective probabilities in (13) and generating a random uniform deviate. Isolation of the deviate between any i and (i þ 1) accumulated probability determines the number of branches. 3.4. Estimation of minor branch sizes Madgwick, 1994 observed that when branches in clusters are ranked by size and expressed relative to the maximum branch (of the cluster) then a plot of average relative branch diameter against branch rank shows a close exponential relationship. We explored this with view to modeling the lesser branches in this way but found the relationship was over-simplified by pooling all the diameters regardless of branch numbers. Fig. 9 shows the relationship when the ranked data are separated into separate strata for branch numbers 3–8. Neither diagram highlights the extreme range of data. Table 6 gives the relative branch statistics for different branch numbers. Table 6 shows evidence of distinct distribution characteristics by branch number, so we decided to
xa1 ð1 xÞb1 ; Bða; bÞ
0
(14)
where a, b are shape parameters; Bða; bÞ ¼ GðaÞGðbÞ= Gða þ bÞ and G the gamma function. A Chi-squared goodness-of-fit statistic was generally non-significant, substantiating a tenable fit. Figs. 10 and 11 show the branch histograms and fitted functions for four and eight branches that represent the worst and best outcomes. The minor branch sizes in any cluster are estimated by first predicting the maximum branch size and total number of branches (Eq. (13)) then addressing a sequence of random deviates at the appropriate accumulative form of (14). 4. Model output 4.1. Summary of model logic The model is addressed through an input file containing a tree list of diameters (at breast height) and corresponding tree heights, and optionally, for any tree, an ocular count of the number of whorls to the GC and a measure of BF, defined in (3) above. The basal area of the top 100 stems per hectare is assumed to be calculable. Output is controlled by an iterative loop, each tree’s branch characteristics being generated consecutively. The following sequence of operations occur for each tree. 1. The mean, minimum, and standard deviation of maximum branch size are calculated through models (2), (4) and (6) or (3), (5) and (7) depending on whether BF measures are available. 2. A mean-tree internode length is assigned through (11). 3. The height to the GC is estimated by (12). 4. A succession of internode length deviations are generated by (11). Each is added to the mean value in (2) to create a internode length, and
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255
Fig. 9. Relationship between branch number and relative branch size.
accumulated until a relative height greater than unity is recorded. Each length is adjusted for height by multiplying by the factor calculated in model (10). 5. A Weibull distribution of maximum branch sizes is created by using the values in (1) and estimating the scale and shape parameters of the Weibull through Garcia’s method. The location parameter is equated to the minimum (maximum) branch, estimated in (1). 6. For each internode length a maximum branch size is assigned through (9). Each branch is adjusted for height (bulge) effects by multiplying by the factor calculated in (1). Maximum branch estimation ceases when GC height in (3) is reached.
7. The estimated number of branches at each whorl is estimated by (13). 8. The sub-branch dimensions for any whorl are generated by choosing the appropriate beta function (14) dependent on the number of branches given by (7). The branch sizes are then created as described in Section 3.4. 4.2. Branch characteristics created for one tree The branch output created by the model for one tree are copious and can be highly variable. We give the results of two simulations for one tree of identical dimensions. Table 7 summarizes these simulations for a tree of diameter breast height of 450 mm, total
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Table 6 Summary statistics for minor branches by branch number Number of branches
Mean
Standard deviation
Minimum
Maximum
Skewness
Kurtosis
(2, 3) 4 5 6 7 8þ
0.62 0.58 0.57 0.53 0.53 0.51
0.22 0.23 0.25 0.25 0.25 0.25
0.11 0.07 0.07 0.06 0.05 0.04
0.98 0.98 0.98 0.99 0.98 0.98
0.43 0.20 0.20 0.04 0.04 0.09
0.77 0.93 1.17 1.16 1.15 1.12
height 30 m and residing in a stand where the top 100 stems is 18 m2/ha at age 20. Fig. 12 shows maximum branch size plotted against cluster height up to the GC, for the two simulations.
4.3. Branch characteristics for various stockings Woollons and Whyte, 1989 describe a P. radiata thinning experiment located in the Central North
Fig. 10. Histogram and fitted beta function of relative branch size for four branches.
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Fig. 11. Histogram and fitted beta function of relative branch size for eight branches.
Table 7 Branch characteristics for two simulations Mean
Standard deviation
Minimum
Maximum
Run 1 Maximum branch (mm) Internode length (m) Number of branches/whorl
42 0.43 4.9
18 0.26 2.4
15 0.06 1
75 1.15 11
Run 2 Maximum branch (mm) Internode length (m) Number of branches/whorl
40 0.50 5.0
17 0.34 2.2
16 0.05 1
84 1.75 10
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Fig. 12. Predicted maximum branch size and relative height for two simulations.
Table 8 Predicted maximum branch size statistics (mm) for various stockings Stocking
Mean branch size
S.D.
Minimum
Maximum
200 300 400 500 600 700 1200
40 35 32 28 27 26 20
17.4 15.9 15.3 13.6 13.0 12.7 11.4
10 8 7 7 7 6 4
117 125 104 102 104 101 90
Island of New Zealand, where seven stocking densities are compared. Using these data Table 8 together with Figs. 13 and 14 summarize the maximum branch size estimates (up to the GC) from the model for different stockings. 5. Discussion and conclusions The model described here represents an attempt to obtain estimates of internode length, branch cluster
number and size for P. radiata in Carter Holt Harvey’s forests in New Zealand. On two counts the model represents a mild departure in methodology compared to other models: (a) the use of the whole tree as the sampling unit rather than the annual shoot; and (b) the part use of simulation techniques to generate internode length and branch characteristics. We have stated our reasons not to model annual shoot growth and in hindsight we have not regretted this decision. The use of stochastic procedures in branch models is certainly not new (see for example, Cochrane and Ford, 1978; Grace et al., 1998) but here we utilize up to five independent sub-models to produce the output data. In each case however, the decision to adopt a stochastic process was only made after rigorous testing that more deterministic relationships were not available. An interesting outcome of modeling maximum branch size is the detection of larger branches at approximately 30–40% of total stem height. For other species this has been observed before (for example, Makinen and Colin, 1998; Petersson, 1998). The occurrence is less pronounced than that found for
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Fig. 13. Average branch size and stand density.
internode length but clearly the effect is worth recognising (and modeling). The fluctuations around 35% may be a function of crown height (caused by different stockings) but it also suggests that larger branches may also be associated with bole growth rate. The phenomenon is coincident with the peak of P. radiata height and diameter increment that occurs around 4–6 years. The models developed to generate the maximum branch distributions are remarkable for the absence of stems per hectare as a predictor variable. Although tried repeatedly in various functional forms there was no justification for its inclusion in any of the models. Many operational foresters may find this result surprising and indeed stocking has been traditionally synonymous with branch size and its control (Smith,
1986). Implicitly however, stocking is associated here with branch size through its close correlation with tree diameter, which is by far the dominant predictor variable in Eqs. (3)–(8). Given that larger diameters are generally associated with lower stockings then the influence becomes clear. The regression coefficient associated with tree height in models (3) and (4) is negative, suggesting that for trees of equivalent diameter but more height will have somewhat smaller branches but the effect is not substantial. The success of the variable-basal area per hectare of the top 100 trees is interpreted as a measure of productivity. The exponential functional form of diameter in Eqs. (5) and (6) is an important result implying that disproportionate variation is encountered with larger trees. In practical terms, a dominant tree is very likely to have
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Fig. 14. Frequency distribution of predicted branch sizes for three densities.
not only large branches but also the biggest range of branch sizes. The lack of evidence for regional and stand differences in internode length is comforting as this permits some use of the model in adjacent (nonsampled) forests without fear of large errors in prediction. The relationship between changes in internode length and stem height has been reported earlier for P. radiata (Grace and Carson, 1993). Beginning slowly but quickly peaking around 30% of relative stem height before subsiding this is also in harmony with P. radiata height and diameter increment development. The correlations in Table 4 confirm that internode length of P. radiata is essentially independent of tree size or site productivity while the absence of consistent autocorrelation between successive branch clusters simplifies the model construction. From the analysis of variance in Table 5 the dominant source of internode variation is attributable to fluctuations along the stem. A second, but appreciably smaller source of variation can be attributable to among-tree differences, probably reflecting distinct genetic origins (Grace and Carson, 1993). Nevertheless, we decided to recognize this source of variation in the model, in
anticipation of encountering trees that are characterized by extremely long internodes (trees so bred are currently younger than 20). The distributions of the minor branches (Table 6) change with increasing branch number from skew to the right to the left with increasing branch number. This reflects a tendency for clusters with few branches to be large in size, while clusters with many branches to be generally small, but the relationships are not especially strong. The summary of the branch characteristics for a single tree (Table 7 and Fig. 12) highlight the high variation in branch size that is encountered for P. radiata while the two simulations give an estimate of the expected phenotypic variation for trees of similar size. The output for the various stockings is interesting on at least two counts. While there is an expected exponential decrease in average branch size with increasing stocking, the degree of overlap in the distributions may surprise some foresters. Thus, while branch size is obviously associated with stems per hectare it is inevitable that a component of logs from any regime will have large branches, or conversely, adoption of a wide spacing will not always result in big branches. Currently the model has some limitations. For example, it cannot be reliably used when trees have been subjected to later age thinnings; there is evidence these regimes can induce changes in crown and branch dimensions (Siemon et al., 1976). Moreover, because maximum branch dimensions are imposed at a standard age of 20, live branch clusters above the GC height currently cannot be estimated reliably. Nevertheless, the functions developed here are sufficient to estimate branch characteristics up to around 40% of total height, thereby catering for by far the most valuable section of the bole.
Acknowledgements The model described in this contribution was developed from the ideas of several people. In particular, we wish to acknowledge the enthusiasm and interest of Mr. Neil Eder, Carter Holt Harvey Forests, without which this project would not have developed. Dr. E. Mason supplied a critical commentary at an earlier stage. We are also grateful to two anonymous referees for their contributions.
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References Bailey, R.L., Dell, T.R., 1973. Quantifying diameter distributions with the Weibull function. For. Sci. 19, 97–104. Ballard, R., 1977. Predicting fertilizer requirements of production forests. Use of Fertilizers in New Zealand Forestry. In: Proceedings of the FRI Symposium No. 19, March 1977, N.Z. For. Serv. Bannister, M.H., 1962. Some variations in the growth pattern of Pinus radiata in New Zealand. N.Z. J. Sci. 5, 342–370. Brazier, J.D., 1977. The effect of forest practices on the quality of the harvested crop. Forest 50, 911–916. Brown, G.S., 1962. Stages in branch development and their relation to pruning. N.Z. J. For. 8, 608–622. Burdon, R.D., 1994. Annual growth stages for height and diameter in Pinus radiata. N.Z. J. For. Sci. 24, 11–17. Carson, M.J., Inglis, C.S., 1988. Genotype and location effects on internode length of Pinus radiata in New Zealand. N.Z. J. For. Sci. 18, 267–279. Chatfield, C., 1985. The Analyses of Time Series, 3rd Edition. Chapman & Hall, London. Cochrane, L.A., Ford, E.D., 1978. Growth of a Sitka spruce plantation: analyses and stochastic description of the development of the branching structure. J. Appl. Ecol. 15, 227–244. Draper, N.R., Smith, H., 1981. Applied Regression Analysis, 2nd Edition. Wiley, New York. Eng, G., Daellenbach, H.G., Whyte, A.D.G., 1986. Bucking treelength stems optimally. Can. J. For. Res. 16, 1030–1035. Fielding, J.M., 1960. Branching and flowering characteristics of Monterey Pine. For. Timb. Bur. Bull. No. 37, Canberra, Australia. Garcia, O., 1981. Simplified method-of-moments estimation for the Weibull distribution (note). N.Z. J. For. Sci. 11, 304–306. Goulding, C.J., 1986. Forestry Handbook. N.Z. Inst. For. Grace, J.C., Carson, M.J., 1993. Prediction of internode length in Pinus radiata stands. N.Z. J. For. Sci. 23, 10–26. Grace, J.C., Blundell, W., Pont, D., 1998. Branch development in Pinus radiata model outline and data collection. N.Z. J. For. Sci. 28, 182–194. Greets, J.M.P., 1984. Mathematical solution for optimizing the sawing pattern of a log given its dimensions and its defect core. N.Z. J. For. Sci. 14, 124–134.
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Inglis, C.S., Cleland, M.R., 1982. Predicting final branch size in thinned radiata pine stands. F. R. I. N.Z. For. Serv. Bull. No. 3. Irwin, J.O., 1959. On the estimation of the mean of a Poisson distribution from a sample with the zero class missing. Biometrics 15, 324–326. Jacobs, M.R., 1938. Notes on pruning Pinus radiata. Part 1. Observations on features which influence pruning. Common. For. Bur. Bull. No. 23, Canberra, Australia. Johnson, N.L., Kotz, S., 1969. Discrete Distributions. Houghton Mifflin, Boston. Law, A.M., Kelton, W.D., 1991. Simulation Modeling and Analysis, 2nd Edition. McGraw-Hill, New York. MacCurdy, E., 1954. The Notebooks of Leonardo da Vinci, Vol. 1. The Reprint Society, London. Madgwick, H.A.I., 1994. Pinus radiata: Biomass, form, and growth. Pers. Publ., Rotorua, New Zealand. Maguire, D.A., Moeur, M., Bennett, W.S., 1994. Models for describing basal diameter and vertical distribution of primary branches in young Douglas-fir. For. Ecol. Manage. 63, 23–55. Makinen, H., Colin, F., 1998. Predicting branch angle and branch diameter of Scots pine from usual tree measurements and stand structural information. Can. J. For. Res. 28, 1686–1696. McEwen, A.D., 1975. PROGRAM/STEMANALYSIS A computer program for detailed stem analysis. N.Z. For. Serv. F. R.I. For. Mens. Manage. Rep. No. 49, unpublished. Petersson, H., 1998. Prediction of branch variables related to timber quality in Pinus sylvestris. Scand. J. For. Res. 13, 21–30. Pnevmaticos, S.M., 1975. Improving recovery from softwood bucking in Eastern Canada. For. Ind. 95, 42–47. Pnevmaticos, S.M., Mann, S.H., 1972. Dynamic programming in tree bucking. For. Prod. J. 22, 26–30. Siemon, G.R., Wood, G.B., Forrest, W.G., 1976. Effects of thinning on crown structure in radiata pine. N.Z. J. For. Sci. 6, 57–66. Smith, D.M., 1986. The Practice of Silviculture, 8th Edition. Wiley, New York. Whiteside, I.D., 1982. Predicting radiata pine gross sawlog values and timber grades from log parameters. N.Z. For. Serv. F. R. I. Res. Inst. Bull. No. 4. Woollons, R.C., Whyte, A.G.D., 1989. Analysis of growth and yield from three Kaingaroa thinning experiments. N.Z. J. For. 34, 17–20.
Modeling internode length and branch characteristics for Pinus radiata in New Zealand R.C. Woollonsa,*, A. Haywoodb, D.C. McNicklec a
School of Forestry, University of Canterbury, Christchurch, Canterbury, New Zealand b Carter Holt Harvey Forests, P.O. Box 648, Tokoroa, New Zealand c Department of Management, University of Canterbury, Canterbury, New Zealand Received 8 October 2000; accepted 2 February 2001
Abstract It is recognized that estimation of internode length and maximum branch size is important for the prediction of clearwood in unpruned timber stands, as well as for evaluating the quality and value of logs in general. A review of existing branch models reveals a diversity of approaches as well as a tendency for results to be species specific. Here, a branch model is developed for Pinus radiata in New Zealand, capable of predicting successive internode lengths, the number of branches on each branch cluster, and the size of each branch up to the green crown (GC) at site index age 20. Inputs to the model include tree height and diameter at breast height (dbh) (both at age 20) and basal area per hectare of the top 100 stems. Further optional inputs are an ocular count of the number of branch clusters up to the green crown, and branch factor (BF)—the size of the biggest branch in the first cluster encountered above 6 m. The vertical distributions of internode length and maximum branch diameter are found to reach maxima around 0.3–0.4 of relative height. Internode length and the number of branches per whorl were found to be independent of tree size, site index and stand density. Stems per hectare is not required as an explicit predictor variable but it appears implicitly through tree dbh. Some model output is given and the results are discussed. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Radiata pine; Branch dimensions; Distance between branch clusters; Predictive simulator
1. Introduction The study of tree branches is certainly not new. Leonardo da Vinci1 (c1500), hypothesized that the sum of branch cross-sectional areas above any branch cluster should be equivalent to the cross-sectional area of the bole below that cluster. For Pinus radiata in Australia, Jacobs, 1938 presented a monograph on branch habits that influenced the efficiency of pruning. * Corresponding author. E-mail address: [email protected] (R.C. Woollons). 1 see MacCurdy (1954).
Fielding, 1960 gave a description of branch morphology for radiata pine around Canberra in Australia, while Bannister, 1962 reported the species in New Zealand, concentrating on annual shoot and branch development from a physiological viewpoint. More recently, several authors have stressed the importance of predicting tree internode length2 and how it may 2 Bannister (1962) points out foresters have developed ambiguous definitions of the terms ‘node’ and ‘internode length’ that can be at variance with physiological and botanical viewpoints. From a modeling perspective, the differences in definition are unimportant and we continue to use the terms. This should not be taken as any rebuttal of Bannister’s stance.
0378-1127/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 1 1 2 7 ( 0 1 ) 0 0 4 6 8 - 6
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vary with breed, site, and management, because it is an important variable in determining amounts of clearwood which can be obtained from unpruned logs (Carson and Inglis, 1988; Grace and Carson, 1993). Many current end-uses for P. radiata clearwood require lengths below 2 m and often less than 0.5 m. At the same time, branch size and number are also well recognized to be vital factors in assessing quality and value of timber (Brazier, 1977; Whiteside, 1982). These points lead to the feasibility of predicting internode length, cluster number, and the sizes of branches thereon by some simulation system. Apart from being a potential asset in evaluating various silvicultural regimes it is seen as a critical tool to enhance the utility of bucking algorithms by providing more information to assess log value (Pnevmaticos and Mann, 1972; Pnevmaticos, 1975; Greets, 1984; Eng et al., 1986). There are numerous branch models described in the literature (see, among many, Cochrane and Ford, 1978; Maguire et al., 1994; Makinen and Colin, 1998) but comparisons between them quickly become complex and diffuse. The response variables, branch angle, branch size, whorl numbers per annual shoot, vary from one study to another as can the sampling units chosen-annual shoot, whole tree, crown, primary or secondary branch. Moreover, results are strongly dependent on species. Maguire et al., 1994 comment ‘. . . model characteristics vary as a result of differences among species. . . specifically in the relative prominence of nodal verses internodal branches’. Whereas P. radiata and Pinus sylvestris, for example form one or more distinct nodes per year with no internodal branches, Chamaecyparis produces branches regularly along the stem and Pseudotsuga is intermediate between these patterns. It follows that useful comparisons between models should be limited to the same species and to models with similar objectives. Two branch models exist for radiata pine in New Zealand. Inglis and Cleland, 1982 built a multiple regression model which predicts branch index defined as the mean of the largest four branches for a nominated log length, taking one from each of four cross-sectional quadrants (regarding the log as a cylinder), at nominated log height classes. Log lengths are required to be between 5 and 6 m. The model uses four stand-level predictor variables; site index (mean-top-height at age
20), mean-stand-diameter at age 20, height class (of nominated logs), and mean-top-height at any (previous) thinning. The model error is calculated as 37 mm with a R2-value of 0.92, although this figure is inflated by the pooling of several thousand data. Grace et al., 1998 discuss the structure of a branch model where the numbers of branch clusters are predicted in relation to annual shoot growth. No actual formulae are given but emphasis is placed on mechanistic principles, and total branch composition and internodal structure are obtained by summing the results of the annual shoot development after growing the branches to maximum size. We reviewed these models and decided that neither suited our objectives. The multiple regression model is somewhat dated and gives virtually no information on either internode length or individual branch size, while the process-based approach assumes that annual shoot length can be satisfactorily and precisely modeled an assumption that seems doubtful considering the complexity of shoot development (Bannister, 1962; Burdon, 1994). We were also influenced by measurement considerations. Adoption of a mechanistic model meant that the data would require the collection of stem analysis material, a complex and time consuming exercise (McEwen, 1975) compounded by a likelihood that shoot discontinuities would be difficult to assess with any certainty (Burdon, 1994). Accordingly it was decided to ignore shoot and branch growth. Instead, for a given tree, we would estimate the number of dead branches comprising any cluster, the respective (final) branch sizes, and the distance between any two consecutive clusters (internode length) up to the green crown (GC). In this contribution we describe the construction of such a P. radiata branch model for New Zealand North Island plantations. The construction and rationale of the component equations, together with the model logic are described and some output from the system is presented. Some limitations and future developments of the model are discussed.
2. Data We wanted the model to include both stand and tree input measures. These required the data to be collected in plots and at a fixed (index) age because although
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lower-stem branches have ceased to grow (maximum size is achieved in around 10 years for P. radiata, Brown, 1962; Grace et al., 1998), stem growth continues to develop. Age 20 was chosen as the index— the age New Zealand foresters use to define site index for P. radiata (Goulding, 1986). Moreover, it represented a logical age to carry out stand inventories being only 4–6 years before clearfall. Data were collected from 191 trees in three Carter Holt Harvey Ltd. (CHH) forests (Mahurangi, Riverhead, and Kinleith) in the North Island of New Zealand. At Mahurangi, a high production forest, two 20-yearold permanent sample plots were clearfelled and 60 trees sampled. At Riverhead, a lesser productive area (Ballard, 1977) a single permanent plot of 43 trees was sampled. At Kinleith a more elaborate sampling strategy was employed. The forest was stratified into four large compartments and in each, 20-year-old stands were selected to secure a range of stockings. Six to eight 1/10 ha plots were established in each selected stand and within each, three trees were extensively measured. Residual trees were only measured for diameter at breast height (dbh) outside bark. No thinning had been carried out in any plot later than age 8.
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All trees measured for branch characteristics were sampled as follows. Before felling, an ocular estimate of the number of branch clusters to the GC was obtained, along with a measure of branch factor (BF), defined as the largest branch in the first whorl occurring above 2 m. After felling and measuring tree dbh and total height, the distances between all branch clusters from the butt to the tip were obtained (at Kinleith, they were measured only to the GC). The distance to the GC height was recorded, defined to be the distance to the first branch cluster containing at least one branch with green needles. For all branch clusters, the maximum branch diameter was measured with calipers, directly adjacent to the stem. For every fifth cluster all branches were measured for diameter, regardless of size. Table 1 summarizes these data.
3. Component models 3.1. Maximum branch size 3.1.1. Changes in vertical distribution A plot of average maximum branch size and relative height class shows a pronounced ‘bulge’ at around
Table 1 Summary of branch, tree and stand data Mean
Standard deviation
Minimum
Maximum
Kinleith Stems per hectare Diameter at breast height (dbh, mm) Tree height (m) Site index (m) Maximum branch (mm)
531 353 25.7 28.7 34
241 136 5.5 3.3 20
260 111 13.2 22.9 2
1020 592 37.0 34.2 118
Mahurangi Stems per hectare Diameter at breast height (dbh, mm) Tree height (m) Site index (m) Maximum branch (mm)
488 388 29.5 32.1 41
10 78 3.1 1.6 18
480 210 20.7 30.1 3
500 529 37.2 33.5 115
Riverhead Stems per hectare Diameter at breast height (dbh, mm) Tree height (m) Site index (m) Maximum branch (mm)
430 329 27.3 29.6 26
0 60 3.3 0 13
430 129 18.5 29.6 1
430 475 32.5 29.6 95
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Fig. 1. Relationships between average maximum branch size and relative height of the tree.
0.3–0.4 of relative height then decreasing thereafter. (There are relatively few data above relative height around 0.6 since these are generally above GC height where the branches are still growing.) The relationships for the three forests are depicted in Fig. 1. A function MAXBRrel ¼
hr þd ða þ bhcr Þ
(1)
where MAXBRrel is the relative maximum branch size, defined to be any maximum branch size on the ith tree, as a proportion of the average (maximum) branch size for the ith tree, where the average is calculated over all whorls, hr the relative height along the tree; a, b, c and d are regression coefficients, estimated by non-linear least-squares, was fitted to the data. All coefficients are significantly different from zero, as determined by asymptotic 95% confidence intervals and the lower bounds of these are a minimum of four deviations from zero. The residual values resulting from (1) are unbiased, when compared with predicted values, relative height and forest.
Fig. 2 shows the cumulative ordered residuals plotted against a normal distribution with zero mean; clearly the overall goodness-of-fit is very good. The first attempts to model this effect with variants of Eq. (1) used absolute rather than relative maximum branch size. It was found that this could generate extraordinary small or even negative branch sizes. Usage of relative branch size removed this problem. 3.1.2. Estimation of maximum branch size From Table 1, maximum branch size is strongly (and logically) related to stem diameter. Fig. 3 shows the distribution of branch sizes, scaled by dividing each by the predicted value of (1) given the appropriate relative height for the three forests. Preliminary analyses of the maximum branch distribution showed it was best modeled by the Weibull distribution c x yc1 xy c f ðxÞ ¼ exp ; x>y s s s (2)
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Fig. 2. Goodness-of-fit of residuals for the branch relative height model.
Fig. 3. Frequency distributions of maximum branch size for the three forests.
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where y, s, and c are location, scale, and shape parameters (s > 0, c > 0). Fitting (2) to distribution data can be achieved by a variety of estimation techniques (Bailey and Dell, 1973). Maximum likelihood estimation of such a shifted distribution can be problematic (Law and Kelton, 1991) so we utilized the method of moments but avoided requiring an iterative solution by adopting an explicit approximation proposed by Garcia, 1981. This requires the estimation of the mean, standard deviation, and minimum scaled maximum branch size. After much analysis, six models were adopted, depending on whether the variable BF, defined above, is incorporated as a predictor variable. The figures in brackets below each equation give the significance level for the hypothesis that any coefficient is equivalent to zero. 1. Mean scaled maximum branch size MEANMAX ¼ b1 d20 þ b2 h20 þ b3 BA100 ðall < 0:0001Þ; R2 ¼ 0:96;
s2 ¼ 47:6
(3)
MEANMAX ¼ b1 d20 þ b2 h20 þ b3 BF R2 ¼ 0:98;
s2 ¼ 30:2
3.2.1. Changes in internode length by latitude There is some evidence in New Zealand that internode length may be a function of latitude and site (Carson and Inglis, 1988; Grace and Carson, 1993), with a tendency for the most southerly plantations to have longer internode lengths. Close examination of these studies reveals that the database on which the conclusions are based is very small (one plot in each of 16 forests throughout the country), and thus, some doubt must exist as to the strength of the relationships claimed. Consequently we decided to ignore this facet of internode behavior in the model. Table 2 and Fig. 4 summarize the internode length data for the three
(4)
ð0:01Þ
ð0:025Þ ð0:001Þ
s2 ¼ 20:6
(5)
STDMAX ¼ b1 expðb2 d20 Þ þ b3 BF ð0:005Þ
ð0:025Þ ð0:01Þ
þ b4 MEAN BF; ð0:025Þ
R ¼ 0:45;
(9)
3.2. Estimation of internode length
STDMAX ¼ b1 expðb2 d20 Þ þ b3 BA100
2
MAXBR ¼ ððlogð1 rÞÞ1=c Þs þ MINMAX
þ b4 MEAN BF ðall < 0:0001Þ; 2. Standard deviation of scaled maximum branch size
R2 ¼ 0:37;
where in (3)–(8) above: (MEAN, STD, MIN) MAX are mean, standard deviation, and minimum scaled maximum branch size; BA100 is basal area per hectare of the top 100 trees; MEAN_BF the average (plot) BF; d20 the dbh, at age 20 (mm); h20 the total stem height, at age 20 (m), and all other terms are defined earlier. The residuals of the six models are without exception, without serious bias. The models are devoid of intercepts to avoid aberrant behavior near zero. To estimate maximum branch sizes for a given tree, appropriate inputs in (3)–(8) (dependent on whether BF has been measured), give estimates of the mean, standard deviation, and minimum (maximum) branch size. Using Garcia’s moment procedure allows direct calculation of scale and shape parameters in (2) (the location parameter being estimated by MINMAX through (7) or (8)) thus creating a unique Weibull distribution for each tree. A maximum branch estimate (MAXBR) is immediately available through the inverse cumulative distribution form of the Weibull after generating a random uniform deviate (r)
2
s ¼ 18:1
(6)
3. Minimum scaled maximum branch size Table 2 Summary of internode lengths (m) by forest
MINMAX ¼ b1 d20 þb2 BA100 ð0:0001Þ 2
R ¼ 0:82;
ð0:0133Þ
2
s ¼ 24:4
(7)
MINMAX ¼ b1 d20 þ b2 MEAN BF ð0:0001Þ
R2 ¼ 0:82;
s2 ¼ 24:2
ð0:0067Þ
(8)
Forest
Average
Standard deviation
Minimum
Maximum
Kinleith Mahurangi Riverhead
0.60 0.55 0.50
0.38 0.28 0.30
0.08 0.04 0.04
2.38 2.30 2.34
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249
Fig. 4. Frequency distributions of internode length for the three forests.
regions. Kinleith is some 300 km south of the other two forests but the difference in average internode length is not statistically significant. 3.2.2. Changes in vertical distribution When the mean-tree internode lengths were plotted against relative height classes (Fig. 5), a clear systematic trend appeared with generally shorter lengths around the butt, but sharply increasing to around 0.30 of tree height, then gradually decreasing thereafter to the tip. (The effect is unclear when individual lengths are graphed against actual relative heights for all trees because of extreme variation.) A function ILrel ¼
hr þd a þ bhcr
(10)
where ILrel is relative internode length defined to be any internode length on the ith tree, as a proportion of the average internode length of the ith tree; hr the relative height along the tree, a, b, c, and d are regression coefficients estimated by non-linear
least-squares, was satisfactorily fitted to these data. All coefficients were significantly different from zero, as assessed by 95% asymptotic confidence intervals and the lower bounds of these are a minimum of three deviations from zero. Residual values resulting from (10) were unbiased, when compared with predicted values, relative height and forest. Fig. 6 shows the cumulative ordered residuals plotted against a normal distribution with zero mean; the overall goodness-offit is satisfactory. As for maximum branch size the use of relative rather than absolute internode length negated the occasional generation of lengths around zero. 3.2.3. Serial correlation of internode lengths We explored possible serial correlation (Draper and Smith, 1981) between successive internode lengths by fitting model (10) individually to each of the 191 trees, then treating each sequence of residual values as a time series. A first order autocorrelation coefficient (Chatfield, 1985) was calculated for each tree and the results compared. Table 3 summarizes these values.
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Fig. 5. Relationships between internode length and relative height of the tree.
From Table 3 it is concluded that no significantly large serial correlations exist between successive internode lengths. Apart from the low average values there is evidence that the higher values for Kinleith in part may be a function of too few internode numbers-these trees being only measured to GC height. 3.2.4. Relations of internode length with growth and space The internode lengths were scaled by dividing by the predicted value of (10) given the appropriate relative height. These were averaged for each tree, and Pearson correlation coefficients were calculated for various tree and stand variables. Table 4 summarizes these statistics. Although some coefficients are significantly different from zero, for practical purposes the correlations are very weak and can be disregarded. To explore any stand effects for internode distance, a nested random effects analysis of variance was applied to the Kinleith forest data. Results are given in Table 5.
The very dominant source of variation is attributable to fluctuations along the stem. A second but appreciably smaller source of variation might be attributable to tree-to-tree differences, probably reflecting distinct genetic composition (Grace and Carson, 1993). 3.2.5. Prediction of internode length From the above analyses it is evident internode length can be regarded as a random phenomenon apart from systematic effects occurring in relation to relative height and perhaps differences occurring with different trees. To simulate internode length on any stem we created two distributions: 1. the distribution of scaled mean-tree internode distances; 2. the distribution of scaled deviations (residuals) around mean-tree values. Both distributions (1) and (2) are asymmetric, being (logically) skewed to the right. Several probability distribution functions were considered to model these distributions beta, gamma, log-normal, and
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251
Fig. 6. Goodness-of-fit of residuals for the internode relative height model.
Weibull, but after comprehensive analyses and applying various goodness-of-fit procedures the lognormal distributions ! 1 ðlogðx yÞ zÞ2 exp f ðxÞ ¼ pffiffiffi ; 2s2 s 2pðx yÞ x>y
(11)
where s, y, and z are estimated shape, scale, and location parameters (s > 0) and (1) and (2) referring Table 3 First order autocorrelation coefficients of residuals for each forest Forest
Average
S.E.
Minimum
Maximum
Kinleith Mahurangi Riverhead
0.240 0.122 0.006
0.035 0.028 0.028
0.27 0.34 0.33
0.56 0.49 0.39
to the mean and deviation distributions, were adopted. Figs. 7 and 8 show the respective distributions and the fitted log-normal probability functions. The fit of the mean-tree internode lengths is very satisfactory with the Anderson–Darling goodness-of-fit statistic being completely non-significant (P < 0:4). The modeling of the deviations of internode lengths is less precise but still reasonable. To produce internode distances for a specific tree the following algorithms are invoked, dependent on whether an ocular estimate of the number of branch clusters to the stem GC is available. 3.2.6. Prediction of internodes with no estimate of whorl numbers available An average tree internode length (AVEL) is generated from the appropriate log-normal distribution (11)
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Table 4 Correlation coefficients for the scaled internode distances Tree diameter at breast height (dbh)
Tree height
Site index
Stems per hectare
Basal area per hectare
0.193
0.125
0.036
0.035
0.151
Table 5 Nested ANOVA for Kinleith internode lengths Source of variation
D.F.
Mean square
Total variation (%)
Sub-forests Plots within sub-forests Trees within plots Internodes within trees
3 26 60 1904
1.904 0.573 0.364 0.127
1.89 1.95 7.75 88.41
above using standard random-variate generation techniques and the log transformation described in Law and Kelton, 1991. Having generated an average tree value successive internode lengths are generated (for a specific tree) by similarly producing internode deviations (Idi) from (2) above. A series of scaled internode lengths (SILi) are then obtained from SILi ¼ AVELi Idi
Fig. 7. Histogram and fitted normal distribution of residuals for the mean-tree branch model.
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253
Fig. 8. Histogram and fitted normal distribution of residuals for the deviations branch model.
This sequence of scaled internode lengths can be converted to actual lengths by multiplying each SILi value by the appropriate relative length value in (10). These are accumulated and the process is repeated until the total internode lengths equal or exceed the (known) height of the tree.
3.2.7. Prediction of internodes with an estimate of ocular count available When an estimate of the number of branch clusters to the GC (NCGC) is available the estimation process above can be improved.
A model of GC height is given by GCh d20 Cratio ¼ ¼ b1 þ b2 exp h20 1000
(12)
where Cratio is the ratio of green crown height (GCh) to total stem height at age 20 (h20); d20 the dbh, at age 20 (mm); b1 the regression coefficients. Model (4) has an R2 ¼ 0:57 with unbiased residuals, both with respect to predicted and d20 values, while both coefficients are significantly different from zero (P < 0:0001). Usage of (12) allows the estimated NCGC to be converted to an estimate of the total number of branch clusters on the stem (NCS) this
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same number of internode lengths are then generated as for (Section 3.2.6) above. In general, the accumulated lengths estimated will not be equal to total tree height. This is achieved by scaling each length by the factor NCS/h20.
create different models for each number. These distributions proved quite difficult to model with several exhibiting small multi-modal characteristics. Ultimately, we fitted a beta density function to the relative diameters to each class.
3.3. Estimation of branch numbers
f ðxÞ ¼
Preliminary analyses showed that branch cluster numbers were independent of tree size, stems per hectare or site index. For our data branch numbers on clusters varied from 1 to 14 with a mode of five. The frequency distribution created was modeled by a truncated Poisson probability distribution (Johnson and Kotz, 1969). Prðx ¼ kÞ ¼ expðlÞð1 expðlÞÞ1 lk =k! k ¼ 1; 2; . . .
(13)
where the parameter l was estimated by an explicit expression given by Irwin, 1959. A Kolmogorov– Smirnov goodness-of-fit test (P < 0:13) substantiated a reasonable fit to the data. An estimate of branch number is obtained by accumulation of the respective probabilities in (13) and generating a random uniform deviate. Isolation of the deviate between any i and (i þ 1) accumulated probability determines the number of branches. 3.4. Estimation of minor branch sizes Madgwick, 1994 observed that when branches in clusters are ranked by size and expressed relative to the maximum branch (of the cluster) then a plot of average relative branch diameter against branch rank shows a close exponential relationship. We explored this with view to modeling the lesser branches in this way but found the relationship was over-simplified by pooling all the diameters regardless of branch numbers. Fig. 9 shows the relationship when the ranked data are separated into separate strata for branch numbers 3–8. Neither diagram highlights the extreme range of data. Table 6 gives the relative branch statistics for different branch numbers. Table 6 shows evidence of distinct distribution characteristics by branch number, so we decided to
xa1 ð1 xÞb1 ; Bða; bÞ
0
(14)
where a, b are shape parameters; Bða; bÞ ¼ GðaÞGðbÞ= Gða þ bÞ and G the gamma function. A Chi-squared goodness-of-fit statistic was generally non-significant, substantiating a tenable fit. Figs. 10 and 11 show the branch histograms and fitted functions for four and eight branches that represent the worst and best outcomes. The minor branch sizes in any cluster are estimated by first predicting the maximum branch size and total number of branches (Eq. (13)) then addressing a sequence of random deviates at the appropriate accumulative form of (14). 4. Model output 4.1. Summary of model logic The model is addressed through an input file containing a tree list of diameters (at breast height) and corresponding tree heights, and optionally, for any tree, an ocular count of the number of whorls to the GC and a measure of BF, defined in (3) above. The basal area of the top 100 stems per hectare is assumed to be calculable. Output is controlled by an iterative loop, each tree’s branch characteristics being generated consecutively. The following sequence of operations occur for each tree. 1. The mean, minimum, and standard deviation of maximum branch size are calculated through models (2), (4) and (6) or (3), (5) and (7) depending on whether BF measures are available. 2. A mean-tree internode length is assigned through (11). 3. The height to the GC is estimated by (12). 4. A succession of internode length deviations are generated by (11). Each is added to the mean value in (2) to create a internode length, and
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Fig. 9. Relationship between branch number and relative branch size.
accumulated until a relative height greater than unity is recorded. Each length is adjusted for height by multiplying by the factor calculated in model (10). 5. A Weibull distribution of maximum branch sizes is created by using the values in (1) and estimating the scale and shape parameters of the Weibull through Garcia’s method. The location parameter is equated to the minimum (maximum) branch, estimated in (1). 6. For each internode length a maximum branch size is assigned through (9). Each branch is adjusted for height (bulge) effects by multiplying by the factor calculated in (1). Maximum branch estimation ceases when GC height in (3) is reached.
7. The estimated number of branches at each whorl is estimated by (13). 8. The sub-branch dimensions for any whorl are generated by choosing the appropriate beta function (14) dependent on the number of branches given by (7). The branch sizes are then created as described in Section 3.4. 4.2. Branch characteristics created for one tree The branch output created by the model for one tree are copious and can be highly variable. We give the results of two simulations for one tree of identical dimensions. Table 7 summarizes these simulations for a tree of diameter breast height of 450 mm, total
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Table 6 Summary statistics for minor branches by branch number Number of branches
Mean
Standard deviation
Minimum
Maximum
Skewness
Kurtosis
(2, 3) 4 5 6 7 8þ
0.62 0.58 0.57 0.53 0.53 0.51
0.22 0.23 0.25 0.25 0.25 0.25
0.11 0.07 0.07 0.06 0.05 0.04
0.98 0.98 0.98 0.99 0.98 0.98
0.43 0.20 0.20 0.04 0.04 0.09
0.77 0.93 1.17 1.16 1.15 1.12
height 30 m and residing in a stand where the top 100 stems is 18 m2/ha at age 20. Fig. 12 shows maximum branch size plotted against cluster height up to the GC, for the two simulations.
4.3. Branch characteristics for various stockings Woollons and Whyte, 1989 describe a P. radiata thinning experiment located in the Central North
Fig. 10. Histogram and fitted beta function of relative branch size for four branches.
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Fig. 11. Histogram and fitted beta function of relative branch size for eight branches.
Table 7 Branch characteristics for two simulations Mean
Standard deviation
Minimum
Maximum
Run 1 Maximum branch (mm) Internode length (m) Number of branches/whorl
42 0.43 4.9
18 0.26 2.4
15 0.06 1
75 1.15 11
Run 2 Maximum branch (mm) Internode length (m) Number of branches/whorl
40 0.50 5.0
17 0.34 2.2
16 0.05 1
84 1.75 10
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Fig. 12. Predicted maximum branch size and relative height for two simulations.
Table 8 Predicted maximum branch size statistics (mm) for various stockings Stocking
Mean branch size
S.D.
Minimum
Maximum
200 300 400 500 600 700 1200
40 35 32 28 27 26 20
17.4 15.9 15.3 13.6 13.0 12.7 11.4
10 8 7 7 7 6 4
117 125 104 102 104 101 90
Island of New Zealand, where seven stocking densities are compared. Using these data Table 8 together with Figs. 13 and 14 summarize the maximum branch size estimates (up to the GC) from the model for different stockings. 5. Discussion and conclusions The model described here represents an attempt to obtain estimates of internode length, branch cluster
number and size for P. radiata in Carter Holt Harvey’s forests in New Zealand. On two counts the model represents a mild departure in methodology compared to other models: (a) the use of the whole tree as the sampling unit rather than the annual shoot; and (b) the part use of simulation techniques to generate internode length and branch characteristics. We have stated our reasons not to model annual shoot growth and in hindsight we have not regretted this decision. The use of stochastic procedures in branch models is certainly not new (see for example, Cochrane and Ford, 1978; Grace et al., 1998) but here we utilize up to five independent sub-models to produce the output data. In each case however, the decision to adopt a stochastic process was only made after rigorous testing that more deterministic relationships were not available. An interesting outcome of modeling maximum branch size is the detection of larger branches at approximately 30–40% of total stem height. For other species this has been observed before (for example, Makinen and Colin, 1998; Petersson, 1998). The occurrence is less pronounced than that found for
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Fig. 13. Average branch size and stand density.
internode length but clearly the effect is worth recognising (and modeling). The fluctuations around 35% may be a function of crown height (caused by different stockings) but it also suggests that larger branches may also be associated with bole growth rate. The phenomenon is coincident with the peak of P. radiata height and diameter increment that occurs around 4–6 years. The models developed to generate the maximum branch distributions are remarkable for the absence of stems per hectare as a predictor variable. Although tried repeatedly in various functional forms there was no justification for its inclusion in any of the models. Many operational foresters may find this result surprising and indeed stocking has been traditionally synonymous with branch size and its control (Smith,
1986). Implicitly however, stocking is associated here with branch size through its close correlation with tree diameter, which is by far the dominant predictor variable in Eqs. (3)–(8). Given that larger diameters are generally associated with lower stockings then the influence becomes clear. The regression coefficient associated with tree height in models (3) and (4) is negative, suggesting that for trees of equivalent diameter but more height will have somewhat smaller branches but the effect is not substantial. The success of the variable-basal area per hectare of the top 100 trees is interpreted as a measure of productivity. The exponential functional form of diameter in Eqs. (5) and (6) is an important result implying that disproportionate variation is encountered with larger trees. In practical terms, a dominant tree is very likely to have
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Fig. 14. Frequency distribution of predicted branch sizes for three densities.
not only large branches but also the biggest range of branch sizes. The lack of evidence for regional and stand differences in internode length is comforting as this permits some use of the model in adjacent (nonsampled) forests without fear of large errors in prediction. The relationship between changes in internode length and stem height has been reported earlier for P. radiata (Grace and Carson, 1993). Beginning slowly but quickly peaking around 30% of relative stem height before subsiding this is also in harmony with P. radiata height and diameter increment development. The correlations in Table 4 confirm that internode length of P. radiata is essentially independent of tree size or site productivity while the absence of consistent autocorrelation between successive branch clusters simplifies the model construction. From the analysis of variance in Table 5 the dominant source of internode variation is attributable to fluctuations along the stem. A second, but appreciably smaller source of variation can be attributable to among-tree differences, probably reflecting distinct genetic origins (Grace and Carson, 1993). Nevertheless, we decided to recognize this source of variation in the model, in
anticipation of encountering trees that are characterized by extremely long internodes (trees so bred are currently younger than 20). The distributions of the minor branches (Table 6) change with increasing branch number from skew to the right to the left with increasing branch number. This reflects a tendency for clusters with few branches to be large in size, while clusters with many branches to be generally small, but the relationships are not especially strong. The summary of the branch characteristics for a single tree (Table 7 and Fig. 12) highlight the high variation in branch size that is encountered for P. radiata while the two simulations give an estimate of the expected phenotypic variation for trees of similar size. The output for the various stockings is interesting on at least two counts. While there is an expected exponential decrease in average branch size with increasing stocking, the degree of overlap in the distributions may surprise some foresters. Thus, while branch size is obviously associated with stems per hectare it is inevitable that a component of logs from any regime will have large branches, or conversely, adoption of a wide spacing will not always result in big branches. Currently the model has some limitations. For example, it cannot be reliably used when trees have been subjected to later age thinnings; there is evidence these regimes can induce changes in crown and branch dimensions (Siemon et al., 1976). Moreover, because maximum branch dimensions are imposed at a standard age of 20, live branch clusters above the GC height currently cannot be estimated reliably. Nevertheless, the functions developed here are sufficient to estimate branch characteristics up to around 40% of total height, thereby catering for by far the most valuable section of the bole.
Acknowledgements The model described in this contribution was developed from the ideas of several people. In particular, we wish to acknowledge the enthusiasm and interest of Mr. Neil Eder, Carter Holt Harvey Forests, without which this project would not have developed. Dr. E. Mason supplied a critical commentary at an earlier stage. We are also grateful to two anonymous referees for their contributions.
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