## Introduction

It has long been recognized that many biological processes show a strong dependence on body size or mass, and can be described with relationships of the form *Y* = *a*_{0}*M*^{d}, where *Y* is some biological process (e.g. metabolic rate), *a*_{0} is a constant, *M* is the body mass and *d* is the scaling exponent (see Niklas 1994). It has been proposed that the fractal-like nature of the underlying branching network, blood vessel networks in animals and crowns and root systems in plants, is responsible for general allometric relationships in plants and animals (Mandelbrot 1983; West, Brown & Enquist 1997, 1999a, b). However, the exact origin of scaling exponents continues to be debated (Kozlowski & Konarzewski 2004; Etienne, Apol & Olff 2006; Reich *et al.* 2006), and possible differences between plants and animals have been pointed out (Mäkelä & Valentine 2006; Reich *et al.* 2006).

West, Brown & Enquist (1999a) studied the allometric exponent *d* using evolutionary optimization arguments in a fractal framework. They searched for the value of *d* that maximizes the metabolic rate of the organism relative to its total body mass, when *d* is defined in terms of the fractal geometry of the plant. Under their assumption that resources driving the metabolic rate are evenly distributed among all active exchange surfaces, the optimum allometry yielded the well-known result of the ‘quarter-power scaling’, that is, that the optimal allometric exponent *d* between exchange surface areas (leaf area in plants; *Y*) and body mass (*M*), takes on the value *d′* = ¾. We distinguish this scaling exponent between leaf area and woody biomass (*d*′) from the scaling of metabolic rate with body mass (*d*), because they are not necessarily the same (Koyama & Kikuzawa 2009; and see later paras). In trees, this result further implies that the fractal dimension of foliage is 3 and the distribution of foliage is volume filling (West, Brown & Enquist 1999a; Mäkelä & Valentine 2006).

There is very limited evidence on the scaling exponent *d* in trees, because total photosynthetic rate is difficult to measure and typically has relied on proxies such as woody growth rate and water use. Nonetheless, these tests have often been shown not to contradict the quarter-power scaling in trees (Enquist, Brown & West 1998; Niklas & Enquist 2001; Meinzer *et al.* 2005; Mäkelä & Valentine 2006). Scaling of respiratory metabolism, however, has rejected ¾ scaling in favour of near isometric scaling with body mass (Reich *et al.* 2006; Peng *et al.*, 2010). However, there is mounting evidence that the fractal dimension of foliage in tree crowns is less than volume filling (Zeide & Gresham 1991; Zeide & Pfeifer 1991; Mäkelä & Sievänen 1992; Ilomäki, Nikinmaa & Mäkelä 2003; Boudon *et al.* 2006). Given that theoretically, the allometric exponent *d* is actually very insensitive to the fractal dimension of foliage, differences between alternative underlying explanations of *d* may not be tractable through empirical model fitting between foliage mass and woody mass. A more stringent test of the theory could be provided by studying the fractal dimension of foliage as such.

A key assumption in the derivation of the quarter-power scaling from evolutionary optimization was that the resources driving the metabolic rate are evenly distributed among all active exchange surfaces. But what if resource capture also relies on the surface area of the body, and not just on the total area of the exchange surfaces? This is the case in tree crowns exposed to light, an essential resource for photosynthetic production. Models of light capture by leaves are based on incident light on the surface of the crown and its gradual extinction towards the inner parts of the crown (Oker-blom, Pukkala & Kuuluvainen 1989; Cescatti 1997; Nilson 1999). Here, the resources are no longer evenly distributed over all exchange surfaces, which interact with each other by shading. In this case, what would be the fractal dimension of the exchange surfaces that maximizes the scaling of the metabolic rate with respect to body mass?

In this study, we analyse the fractal dimension of foliage and wood in tree crowns theoretically and empirically, with the hypothesis that the fractal dimension of foliage is less than 3. In the theoretical part, we revisit the derivation of the optimum fractal scaling exponent *d* (West, Brown & Enquist 1999b) in the situation where light extinction is allowed to influence the metabolic rate of foliage in crowns. Here, we apply a recent summary model of light interception and photosynthesis in forest canopies that consist of individual crowns with specified dimensions (Duursma & Mäkelä 2007; Sinoquet *et al.* 2007). In the empirical part, we test our theoretical results by compiling a large data set on foliage biomass, branch biomass, crown length and crown width of 17 gymnosperm tree species, spanning a wide range in size from saplings to old-growth trees. Finally, we discuss the theoretical and practical significance of the results.