A multiple scaled fractal tree

J. theor. BioL (1990) 145, 199-206

A Multiple Scaled Fractai Tree JOHN W. CRAWFORD AND IAIN M. YOUNG Soil-Plant Dynamics Group, Scottish Crop Research Institute, Invergowrie, Dundee DD1 5DA, U.K. (Received on 14 September 1989, Accepted in revised form on 9 March 1990) In a study of two species of oak tree, the distribution and arrangement of branch lengths are found to be governed by a simple algorithm. The algorithm which has its footing in a type of fractal self-similarity observed in other physiological structures (West et al., 1986. J. appl. Physiol. 60, 189-197; Gotdberger & West, 1987. Yale J. bioL Med. 60, 421-435), reproduces the observed power-law behaviour of mean branch length with order. Furthermore, the high degree of intra-order variability is accounted for as a natural consequence of the superposition of a multiplicity of scales in the structure. The conclusions of this study point to a genetic rather than environmental origin for the design.

I. Introduction The study o f physiological branching structures has evolved from geomorphic analyses of river networks, dating from the turn of the century. Although this is an unnatural development, since in contrast to river networks many physiological systems such as the bronchial tree, arterial networks and botanical trees are optimized for bi-directional flow, many of these methods have seen considerable success in describing the geometry of complex biological structures (MacDonald, 1983; Fitter, 1982; Leopold, 1971; Weibel & Gomez, 1962). To begin the analysis, segments which connect adjacent nodes are grouped in a hierarchy of orders, to which each is assigned an integer label. Once the ordering hierarchy is established, simple relationships are sought between order number and mean length, diameter a n d / o r number o f segments within a given order. There exist a number of different methods for classifying branching systems. Here we adopt the Weibel ordering scheme (Weibel, 1963) where the order number of a particular branch is simply given by the number o f nodes between the branch and the base of the trunk. This method is preferable for our purposes to the traditional schemes (Horton, 1945; Strahler, 1953) where a bias is introduced by the driving philosophy to group together morphologically similar branches under the a priori assumption that they have the same functional status in the network. Furthermore these methods produce a far smaller data set and the final assignment can be radically altered by the omission of broken or self-pruned branch complexes. There have been attempts to relate the geometry o f botanical trees to adaptive strategies which maximize light interception (e.g. Honda et al., 1981; Borchert & Tomlinson, 1984), optimize mechanical stability (McMahon & Kronauer, 1976) or do both simultaneously (Ellison & Niklas, 1988). Environmental effects appear to have an important influence on branch angle (Fisher & Honda, 1979b), and on 199

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self-pruning induced by self-shading (Horn, 1971). However, it has been suggested (McMahon & Kronauer, 1976; Fisher & Honda, 1979a, b) that structural considerations are more likely to be important than light gathering capabilities in determining one aspect of branching geometry: the internode length ratio. Indeed the existence within a given specimen of a single length ratio between successive branch points is largely taken for granted in many theoretical (Honda, 1971; Honda et al., 1981; Fisher & Honda, 1979a) as well as quantitative (Ellison & Niklas, 1988; Fisher & Honda, 1979b) studies. Empirical evidence (e.g. Leopold, 1971) is usually cited in support of this assumption; however, in conflict with this there is also evidence for significant intra-order variability of branch length (Barker et al., 1973). Most models of tree structure adopt an empirical rather than mechanistic approach, often making the derived principles species specific and resulting in a rather more descriptive than informative picture. The mechanistic viewpoint presented here, however, has a range in applicability across a number of diverse physiological systems and is suggestive of some basic principle of organization and evolutionary stability common to many complex biological structures. 2. Data Collection

Measurements were made on two species of oak tree Quercus robur and Q. petraea. Both trees were situated on the edge of wooded areas. Quercus petraea was growing on the banks of a small stream, with trees adjacent to all but its south-east facing aspect, whereas Q. robur had its unshaded side facing south. The measurements of Q. petraea and the lower parts of Q. robur were made on the unshaded sides of the trees, whereas the upper part of Q. robur was unshaded on all sides. All measurements were taken during winter, when the absence of leaves aided the retrieval of data and there was negligible growth. Nodes defined by dead branches and stumps were included in the measurements, although branch tips were not, as it was assumed that further extension was possible during the following year. 3. Theoretical Implications

Since branch length in any order shows significant variability using any ordering scheme, the ratio q, of lengths between successive generations, must also be allowed to assume a distribution of values. We therefore allow q to vary according to some probability distribution f ( q ) with finite first moment. There then exists two possibilities for the scaling mechanism whereby branch lengths decrease from node to node. A Markov-type dependence of branch length on the length in the previous order leads to the prediction that for order z > 0, the expectation value for the length should depend on z according to (l(z)) = lo(q) ~ = to e - ~ ,

(1)

where 10 is the length in the 0th order, (q) is the expectation value for the internode length ratio and a = -In (q) is the corresponding scale factor which is greater than zero if length decreases with order on average. The observation of an apparent exponential dependence is reported in the literature (Barker et al., 1973) for the

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limited range in z provided by traditional ordering schemes. Alternatively the length o f a succeeding branch may be independent of the length of the parent. In this situation branch lengths scale relative to some internal reference length / with a scale factor a. The lengths o f the branches within a given order z, are then given by ~: e -~z where, instead of a single scale factor operating, there are a number of possible values, each occurring with a frequency given by p(a). The expectation value for branch length at order z is then given by,

(t(z)> = J-p(~)g e - ~ da,

(2)

where the integration is over the relevant range of scales. Therefore tile length of a particular branch depends only on C, the value o f the scale factor and the number of nodes between the branch and the base of the trunk. Clearly, the solution depends on the form o f p(a). In his study o f branching in the bronchial tree, West (1987) derived N

p(a)

ot~,

(3)

with N a n d / x constants, by maximizing the information measure of p(a) (Shannon & Weaver, 1963) subject to the constraint that (ln a) exists. This function obeys the scaling relation

p(fla) = fl-"p(o~),

(4)

where /3 is some constant. Therefore the function which dictates the relative frequency with which each scale occurs in the structure is a fractal of dimension D =/z (Mandelbrot, 1983). The essence o f this type of scaling is that a very small amount of coded information can be expanded to completely determine the behaviour o f p ( a ) (West & Salk, 1987) even although values o f a may range over several orders of magnitude. The fractal dimension o f the distribution of scales p(a), has an essential beating on the overall appearance of the tree. Equations (3) and (4) imply that the distribution of mean branch length with order is itself fractal with dimension 1 - / z , i.e.

(t(z))=/3'-~(t(/3z)>,

(5)

and the solution to this equation may be written in the general form of the expression given in West et al. (1986),

Ao+a, (/(z)> -

In z cos [~21ri-~+ z,_~,

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,

(6)

where Ao, A~ and 8 are constants which encompass l, N and the lower and upper bounds on the distribution o f scale factors. Therefore the fractal property of p(a) implies that branch length should decrease with z on average as a type of harmonically modulated power law. The fractal dimension ~ o f the distribution of scales dictates the rate at which (/(z)) decreases with z and therefore has an important

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bearing on both the structural stability of the tree as well as its overall appearance. For example, if 1 - / z is large then the branch pattern will be "bushier" in comparison to a more open, "broom-like" appearance if 1 - / z is small. 4. Discussion and Conclusions

For small values o f the exponent, power-law and exponential distributions are essentially indistinguishable. It is therefore not possible to differentiate between the two scaling mechanisms with the limited number o f orders returned by the traditional ordering schemes. The data plotted in Fig. 1 illustrates how for the sufficiently large values of the exponent allowed by the Weibel scheme, the observed behaviour of (l(z)) deviates from the traditional exponential (broken line) to a power law (solid curve), indicating the appropriateness of the new fractal model. Curves of the form (6) were fitted to the data by first finding a best-fit power law to estimate A0 and /z. A1,/3 and ~ were then estimated visually and a cube of adjacent values centred on these estimates was searched to find the combination which minimized the sum of the squared errors. This process was repeated until a

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z FIG. 1. Data for the lower part o f Quercus robur, ordered according to the Weibel ordering scheme. The broken line is a fit o f the form given in eqn (1) to the first seven orders, the usual maximum number returned by the Horton (1945) or Strahler (1953) ordering schemes applied to entire trees. The solid curve is a simple power law o f the form I ( z ) ~ z -2"4t a n d shows the deviation for high orders from the traditional exponential law predicted by the new fractal model.

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local m i n i m u m was found. Because o f the nature o f the distribution function p(a), s t a n d a r d G a u s s i a n statistics are i n a p p r o p r i a t e for this analysis. In particular the s t a n d a r d deviation a b o u t the m e a n length in a particular generation d e p e n d s on the u p p e r a n d lower cut-offs in the scale factor t~. A l t h o u g h these are implicit in the fitting parameters, the m e t h o d o f solution to eqn (2) c a n n o t lead to an explicit determination o f their values. The predicted h a r m o n i c signature is immediately a p p a r e n t from the data corres p o n d i n g to Q. petraea s h o w n in Fig. 2, where a curve o f the form (6) is fitted as described above. The results for the u p p e r and lower parts o f the Q. robur are s h o w n in Fig. 3, and a h a r m o n i c a l l y m o d u l a t e d p o w e r law is again substantiated. The parameters o f the fits s h o w n in Figs 2 and 3 are given in Table 1. Interestingly, a simple r e n o r m a l i z a t i o n o f the fit to the data c o r r e s p o n d i n g to the lower part gives a satisfactory fit to the d a t a for the u p p e r part; in particular the same p o w e r law slope applies. Since the slope is determined by p(a), this strongly suggests that the factors which govern the frequency o f occurrence o f each scale in the tree are determined genetically rather than by environment. The conclusion that b r a n c h

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Inz FIG. 2. Data (A) corresponding to Quercuspetraea ordered according to the Weibel schemeis plotted

as In [mean length (cm)] vs. In(order number). The solid curve of the form described in eqn (6) is fitted using a local error minimization procedure and shows the harmonic modulation about the dominant power-law behaviour (- - -). As implied by the form ofp(tr) [eqn (3)] the standard errors on the indicated mean values depend on the unknown upper and lower cut-otis in the scale factor a.

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FIG. 3. As Fig, 2 except relevant to Quercus robur, with the curves a and b corresponding to the lower and upper parts of the tree respectively.

TABLE 1 Parameters of the best fit curves of the form (6) shown in Figs 2 and 3 Tree

Quercus petraea Q. robur (lower) Q. robur (upper)

A 0 (cm)

A I (cm)

1-/z

/3

8

148 3442 27 536

44 1137 9096

0.93 2.41 2.41

3 2 2

0.75 5.57 5.57

length ratios are not directly related to environment is supported by other work (McMahon & Kronauer, 1976; Fisher & Honda, 1979a, b). The environment could still play a role in modifying the structure via pruning induced by light shortage due to shading (Honda et al., 1981), but this would affect branch number density rather than branch lengths provided a recognizable node remained. The renormalization is consistent with an increase in ~' with height up the trunk, dictating the overall shape of the crown and allowing more light to penetrate to the lower levels.

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Comparison of Fig. 3 with Fig. 2 shows that there is no apparent similarity of fit between the two trees. Since the correspondence between the structure of the upper and lower parts of Q. robur suggests that the above ground environment does not influence the scaling mechanism, the dissimilarity between the two trees may arise from the genetic differences between the two species. This is further evidence that the frequency of occurrence of particular scales in the structure is controlled genetically. The results presented here show that the new fractal model can reproduce the observed harmonically modulated power law behaviour of mean branch length as a function o f order, and account for the large intra-order variability as a natural consequence of the contribution from a large range in scales to the structure. In contrast to the situation where a single scale operates, the generation of structure by the superposition o f a large range of scales, each occurring with a different genetically determined probability, involves a degree of redundancy in the coded information describing p(a). Since redundancy allows the suppression o f errors which occur during the transmission of information (Shannon & Weaver, 1963), the observed complexity in the structure of tree branching may therefore be an important requirement of evolutionary stability (West & Salk, 1987). It is not clear how, if at all, the structure o f the tree will evolve with time. Backsprouting which occurs in response to stress or following pruning (Horn, 1971) will have an important impact on the structure if it occurs in branches of low order. Since the trees we examined had not been pruned and showed no obvious sign of the effects of stress, backsprouting is probably not an important consideration here. It is common however, for a branch to grow for several years before producing a daughter branch (Kramer & Kozlowski, 1960). If this time lag is a significant fraction o f the age of the tree, then the structure presently observed could change significantly with time. Finally, nodes remaining after branches are lost through self-pruning will eventually become unrecognizable as such, and will therefore not be included in the analysis. This will be an especially important effect on the trunk. A long term monitoring programme would be required to look for time evolution of the structure. Finally, although the results are compelling, it is clear that more data is needed to determine the generality o f the behaviour. Since data collection is laborious and time consuming, reanalysis of existing data would be more appropriate provided that the identification of nodes is carded out in the same fashion as here. Work is currently underway to determine if root systems are governed by the same mechanism. We are grateful to the Forestry Commission for access to Tentsmuir forest, and to the Royal Botanical Gardens in Edinburgh for identification of the trees. REFERENCES BARKER, S. B., CUMMING, O. • HORSFIELD, K. (1973). J. theor. Biol. 40, 33-43. BORCHERT, R. & TOMLINSON, P. B. (1984). Am. J. Bol. 71, 958-969. ELLISON, A. M. & NIKLAS, K. J. (1988). Am. J. Bot. 75, 501-5t2. FISHER, J. B. & HONDA, H. (1979a). Am. J. Bot. 66, 633-644.

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