1. West et al. [Science, 284 (1999) 1677] derived an optimal body-size scaling exponent under the assumption that resources are evenly distributed among exchange surfaces, leading to the well-known ¾ scaling rule. In trees, this implies a volume-filling branching network (a fractal dimension of 3 for foliage). However, there is evidence that the fractal dimension is less than 3 in trees.
2. Here, we include self-shading in the derivation of optimal fractal dimensions. With self-shading, resources are not evenly distributed among leaves because light enters the crown at the surface and is gradually attenuated within the crown. We find that the optimal fractal dimension can take values between 2 and 3, depending on light interception properties and crown size.
3. For a large data set on foliage and woody biomass in gymnosperm trees, we confirm that the fractal dimension of foliage is less than 3, and that it shows a weak dependence on crown size. However, foliage biomass scaled with crown woody biomass with an exponent of 0·78, very close to the theoretical expectation of ¾ scaling. This can be explained by a deviation from the theoretical prediction in the scaling of crown woody biomass and crown length.
4. Overall, these results confirm a deviation from volume filling in gymnosperm trees, and we provide an explanation for this deviation in terms of optimal metabolic scaling. Because ¾ scaling of foliage biomass is still approximately valid, this implies that metabolic scaling exponents may not be as tightly linked to the fractal dimension of foliage as previously assumed.
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 = a0Md, where Y is some biological process (e.g. metabolic rate), a0 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.
West, Brown & Enquist (1999a) derive a general allometric relationship between metabolic area and body mass through fractal considerations, yielding an allometric relationship between exchange surface areas (leaf area in plants; Y) and body mass (M) with the scaling exponent
where 2 + εa is the fractal dimension of the exchange surfaces (z ≡ 2 + εa takes values between 2 and 3, so that εa is an arbitrary exponent between 0 and 1) and 3 + εa + εl is the fractal dimension of body mass, consisting of 2 + εa and 1 + εl, the latter being the fractal dimension of the network length. The quarter-power scaling law follows from this general expression through the requirement that d is maximum with respect to εa∈ [0,1] and εl∈ [0,1]. The maximum is achieved at εl = 0 and εa = 1 in eqn 1, yielding d = ¾ and z =3 (a volume-filling placement of foliage in crowns). This derivation does not depend on details of the branching structure, but is a result of general geometrical considerations of a fractal-like branching structure (West, Brown & Enquist 1999a).
Following West, Brown & Enquist (1999a), we make a distinction between the interior fractal-like network of an organism (for plants, branches that connect to leaves) and the exterior shape that contains the network (the crown; see also West, Enquist & Brown 2009). The interior network is fractal-like, with N branching levels starting at the crown base (we here ignore scaling of the stem below the live crown; see Mäkelä & Valentine 2006) and ending at the leaves. The number of leaves is proportional to the number of endpoints in the network, and sums to the total leaf area (AL). The woody mass connecting to the foliage has a mass of MB (biomass of wood in the crown) and a volume of VB (the volume of wood in the crown), and we assume that MB ∝ VB (constant wood density). The external surface of the crown (including the empty space) has an area AC, a volume VC and a length LC. Because the internal network is fractal-like, we can relate leaf area to crown length as: , where εa is an arbitrary exponent ∈ [0,1], and εa = 1 gives a ‘volume-filling’ branching structure so that leaf area scales with crown volume. Scaling with crown surface area gives because AC ∝ LC2 if crown shape is constant. It can be shown that, in a fractal-like crown, (West, Brown & Enquist 1997, 1999a,b; Mäkelä & Valentine 2006), so that woody biomass in the crown scales with more than the fractal dimension (z =2 + εa).
Although precise estimation of light interception by single crowns is complex, recently two studies have independently shown that time-integrated interception can be well approximated with a simple expression. Both Duursma & Mäkelä (2007) and Sinoquet et al. (2007) used simulations with detailed models (Oker-blom, Pukkala & Kuuluvainen 1989; Cescatti 1997; Nilson 1999) as a basis for developing simplified equations, and both have proposed equivalent summary models as a result. According to these, the radiation intercepted (I) by a crown of envelope surface area AC and foliage area AL is:
where k is an extinction coefficient that depends on leaf clumping and leaf angle distribution. The crown envelope surface area is the area of a convex hull spanning the crown (i.e. the area of a cloth draped over the crown). This relationship was found to hold for long narrow crowns as well as relatively flat crowns when holding AC constant (Duursma & Mäkelä 2007), so that AC and AL alone explain well the interception of light by single trees summed over the growing season.
We make the following assumptions:
1. The fractal dimension of the mean path length from crown base to leaf (l) is 1, that is, εl = 0 (and hence l ∝ LC). This value is independent of AL and metabolic rate and maximizes the scaling of foliage mass and metabolic rate with respect to woody mass in crown (West, Brown & Enquist 1999a).
2. Annual photosynthetic production is proportional to annual intercepted light.
Using these assumptions and eqn 2, we can express the annual photosynthetic rate (P) as:
Next, we define a scaling exponent x as
Clearly, x is not constant but varies with crown size and εa through eqns 3 and 4. Because , and thus , P scales with MB as:
The metabolic scaling exponent β can now be found as:
It is only at the limit kAL/AC → 0 that P scales with foliage area AL. In this case, the exponential term in eqn 3 may be approximated by linearizing it in the neighbourhood of 0, which yields
This means that for small kAL/AC, maximizing the metabolic scaling reduces to the maximization of foliage area scaling, hence yielding the quarter-power scaling (cf. West, Brown & Enquist 1999a). This situation occurs if either (i) the canopy is very sparse (small AL/AC ratio) or (ii) the light extinction is very slow, for example, as a result of within-canopy clumping of foliage (small k).
At the opposite extreme when kAL/AC is very large, P becomes approximately proportional to crown surface area independently of the scaling exponent εa, so that P ∝ AC. In this case, the maximum of β (eqn 7) is obtained with εa= 0.
This analysis suggests that the light interception properties of tree crowns largely determine the optimum metabolic scaling. Furthermore, there is evidence that those properties are variable not only between species but possibly also during the lifetime of trees (Valladares & Niinemets 2007). In reality, neither of the aforementioned extreme cases of crown structure (absence of self-shading or infinitely dense crowns) can be found. It therefore seems likely that if optimal metabolic scaling exists, the scaling exponent εa should lie somewhere between the extreme cases, that is, 0 < εa < 1. Next, we will test this hypothesis against data.
Materials and methods
We compiled a large data set based on destructive harvest of 17 species of gymnosperm trees from a wide range of sites in the northern hemisphere. Trees ranged in size from saplings to large mature trees (height ranged from 0·5 to 45 m and stem diameter from 1 to 98·5 cm).
We compiled a data set of crown size (height, crown length, crown diameter), foliage biomass (MF) and branch biomass (MB) from published and unpublished sources. In all cases, MF and MB were measured by destructive harvest. For smaller trees all foliage was usually measured by weighing, whereas for larger trees individual branches were sampled and scaled to crown totals. Methods for this whole crown estimation varied, but give very good approximations to weighing the whole crown (Monserud & Marshall 1999). The primary criterion for inclusion of a data set was that a sufficiently large range in size was available from trees growing on similar sites. For species-specific scaling, we treated separate studies on the same species as distinct data sets, and in one case split the data set further to account for large variation in site fertility (see Table S1 in Supporting Information). Variation in tree size within a data set originated both from within- and across-stand variation. Both sources of variation showed very similar scaling exponents (see also Mäkelä & Valentine 2006).
Crown envelope surface area and data analysis
Because crown envelope surface area (AC) is difficult to measure, we used the fact that AC is proportional to the product of crown length (LC) and crown width (LB) when the overall shape of the crown profile is constant. We tested whether AC ∝ LCLB for a data set on three gymnosperm species where individual branch lengths were measured (Monserud & Marshall 1999). We estimated the crown surface area based on these branch lengths for crowns ranging in length from 2·5 to 32·9 m, and found for each species that this more accurate estimate of AC was directly proportional to LCLB [the slope in a regression of log(AC) on log(LCLB) was not different from unity at α = 0·05; results not shown]. As the goal in this study was not to provide estimates of actual AC or AL/AC, but rather to study scaling of AL with AC, LCLB can be used as a proxy for AC.
Similarly, we used foliage mass (MF) as a proxy for AL, assuming that AL ∝ MF, but that the proportionality constant (specific leaf area) could vary between species. It is not clear whether AL scales isometrically with MF because shaded foliage has a higher specific leaf area (AL/MF; Niinemets 1997), and it can perhaps be expected that the proportion of shaded foliage within the crown changes with crown size. We tested this assumption of size-invariant specific leaf area on a subset of the data where foliage area (AL) was measured for each sample tree by subsampling the weighed foliage (MF) and measuring projected or total surface area. Foliage area was expressed as an all-sided surface area using species-specific conversions (Barclay & Goodman 2000; Sellin 2000). From the whole data set, 13 species–site combinations were available to test the scaling of AL with MF (see Fig. 1), and we found that scaling exponents were very close to 1·0. We did find that many of the exponents were significantly different from 1·0 (see Table S2), but differences from 1·0 were small. We concluded that MF can be used as a substitute for AL, but recognize the large interspecific variation in specific leaf area (Reich, Walters & Ellsworth 1997).
The allometric scaling of body mass applies to woody biomass in the crown, but only branch biomass was available for all data sets. For a number of studies, stem biomass was also measured but it includes the stem below the crown, which we do not include in our scaling model (see Mäkelä & Valentine 2006). On a subset of the data for which crown stem biomass could be estimated, we found that branch biomass was proportional to total crown woody biomass, so that scaling exponents of branch biomass alone can be used to test model predictions.
If tree crowns were simple geometric solids with constant foliage density, we would expect that foliage area scales with crown volume, or as LC3. In real trees, however, this exponent (z) can assume a range of values between 2 and 3. Here, we assume that tree crowns consist of self-similar branching networks (cf. Mandelbrot 1983; West, Brown & Enquist 1999a; Mäkelä & Valentine 2006). In this framework, the exponent z is called the fractal dimension (Zeide & Pfeifer 1991), and summarizes the distribution of foliage within crowns. The nature of the branching network is in principle responsible for variation in fractal dimensions of real crowns (Mandelbrot 1983). However, instead of measuring branching parameters, it is possible to estimate z from regression of leaf mass on crown size (on a log–log scale) across a wide range in crown sizes (Zeide & Pfeifer 1991; Boudon et al. 2006). Thus, estimates of z and εa were obtained from the data by regression of MF on AC for the whole data set, as well as for each species–data set combination separately, using the relationship log(MF) = w0 +(z/2) log(AC) and εa = z−2. This method to estimate the fractal dimension of foliage (Zeide & Pfeifer 1991) yields a single estimate of z for each data set, assuming that the relationship is linear on a log–log scale. We also tested if estimates of z depend on AC itself, which would result in a nonlinear dependence of MF on AC on the log–log scale. From the data, we also estimated εa as a function of AC by fitting the curve, log(MF) = w0 +w1 log(AC) + w2 log2(AC), and calculating z/2 as dlog(MF)/dlog(AC), which yields z/2 = w1 +2 w2AC. Note that for this quadratic fit, we did not use standardized major axis regression, but simple linear regression, so that the estimates of z are not directly comparable between the two fits (Warton et al. 2006).
To estimate the fractal dimension of foliage (z), we fit the relationship to the whole data set on a log–log scale, and to each species–site combination separately. The exponent εa is found as z−2. For all data combined, we found that z =2·64 [95% confidence interval (CI): 2·60–2·68; and hence εa = 0·64], and slope estimates for all studies separately were concentrated around this mean value (Fig. 2), but with substantial variation between species. Of the 27 exponents estimated for different species–site combinations, eighteen were significantly smaller than 3, eight were not different from 3 and one was larger than 3 (see Table S1). We therefore found no evidence for either surface- or volume-scaling of foliage in tree crowns, but instead found an intermediate value to be more general.
To test if estimates of z depend on body size itself, we fit a quadratic curve to the whole data set, from which we estimated z/2 as the local slope of the curve. We found that the quadratic term in the regression of MF on AC was significant (P < 0·01), but R2 increased by only a fraction (<1%). Estimates of z from this quadratic were found to decrease with body size (Fig. 3), so that foliage in small crowns scaled more closely with crown volume than in large crowns.
For branch biomass (MB), we found that MB scaled with AC with exponent 1·64 (95% CI: 1·62–1·67) for all data combined (Fig. 4). There was, however, considerable interspecific variation in this scaling exponent; ranging from 1·17 to 2·07. Of the 25 species–site combinations (two data sets did not report branch biomass), 11 scaling exponents were not different from 1·5, whereas 7 were significantly smaller than, and 7 larger than 1·5. These results imply that MB scales nearly isometrically with crown volume, which was confirmed by a regression of MB on crown volume (scaling exponent 1·11, 95% CI: 1·09–1·13). Finally, we tested the scaling of foliage biomass with branch biomass (MB ∝ MFd′), and found that d′ = 0·78 (95% CI: 0·77–0·79; Fig. 5). This value is close to the expected ¾ scaling exponent, although significantly different.
West, Brown & Enquist (1999a) showed that the assumption of an even distribution of resources among active surface areas (and hence metabolic rate proportional to total active surface area) leads to a volume-filling branching network that maximizes metabolic rate at a given body mass. In this study, we argued that the distribution of resources among active surfaces (foliage) cannot be assumed even in tree crowns where self-shading is a focal determinant of resource capture. Self-shading has the effect that, at a given body size, metabolic rate does not scale isometrically with leaf area because additional leaf area reduces metabolic rate of the existing foliage. We showed that if this is taken into account the fractal dimension of foliage (2 + εa) varies between 2 and 3, depending on the degree of self-shading in crowns. In very sparse crowns (when AL/AC tends to zero), the model is consistent with that of West, Brown & Enquist (1999a), whereas with large AL/AC the scaling of metabolic rate becomes proportional to crown surface area. We estimated the fractal dimension from a large data set on gymnosperm tree species, and found that it supported an intermediate value between surface area and volume scaling.
Other studies are consistent with our result that, for trees, the fractal dimension of foliage is less than 3, so that branching networks are not volume filling (Zeide & Gresham 1991; Zeide & Pfeifer 1991; Ilomäki, Nikinmaa & Mäkelä 2003; Boudon et al. 2006). This finding is consistent with the formation of an empty space inside crowns owing to twig senescence (Jack & Long 1992). In animals, however, a volume-filling supply network makes more intuitive sense: blood delivers nutrients to all parts of the body, because all body parts are metabolically active. In plants, however, the branching network does not need to be volume filling, because the supply of light, an important resource for photosynthetic production is only to the outside surface of the crown. Self-shading relates to individual crowns and is essentially different from competition for light between individuals which has already been used as a basis for community scaling rules (Enquist & Niklas 2001; Muller-Landau et al. 2006; West, Enquist & Brown 2009).
As predicted by the fractal scaling model, MB scales with the effective crown length (AC1/2) with an exponent larger than the fractal dimension (3·3 as compared with z =2·64). However, the model of West, Brown & Enquist (1999a), with the assumption that εl = 0, predicts the quarter-power scaling with z =3, MB ∝ LC4, MF ∝ MB3/4 (because MF ∝L3 and MB ∝ L4) and hence the allometric scaling exponent d′ = 0·75. From our data set we found an exponent of 0·78, which is very close to the expected 0·75 although significantly different (Fig. 5). However, earlier research has pointed out that the result of a fairly stable exponent d′ only applies to crowns (excluding the branchless bole below the live crown) and not to entire trees, where d′ has been found to vary widely according to crown ratio (Mäkelä & Valentine 2006). This result is not consistent with the model of West, Brown & Enquist (1999a), which predicts that the scaling of foliage biomass with crown woody biomass is related to the fractal dimension (through eqn 1). We found that z ≈ 2·6, which yields d′ = 0·72 from eqn 1 under the assumption that εl = 0. The deviation of the branch scaling exponent explains this discrepancy with the value d′ = 0·78 from the data: we found that branch biomass scaled with a smaller exponent than predicted (3·3 vs. 4) by the model of West, Brown & Enquist (1999a), which results in foliage mass increasing somewhat faster with body mass than expected (because body biomass increases somewhat less than expected).
Based on the simplified light interception model (eqn 2), a fractal dimension of z > 2 implies that self-shading increases with size in gymnosperm trees (because AL/AC ∝ ACz/2−1). If photosynthetic production is proportional to intercepted light, it can be expected that production per unit leaf area decreases with size. For trees, reduced production per unit leaf area with increased size is commonly observed (see review by Ryan, Phillips & Bond 2006), but is attributed to hydraulic constraints or respiratory loads, or other yet unrecognized mechanisms related to size (Mencuccini et al. 2007). Increased self-shading may be an additional factor that leads to reduced growth per unit leaf area (Niinemets, Sparrow & Cescatti 2005).
A number of compensating mechanisms to increase self-shading do need to be mentioned that may make production more proportional to leaf mass than may be expected if production is proportional to light interception. These mechanisms include changes in light availability and distribution in older stands (e.g. Parker, Davis & Chapotin 2002), physiological and structural acclimation of foliage to light (e.g. Niinemets 1997) and nutrient availability (Gower, Vogt & Grier 1992), within-canopy variation in shoot structure (Stenberg 1998) and that light-use efficiency is not constant but changes according to within-canopy distribution of light, and hence degree of self-shading (Oker-blom, Pukkala & Kuuluvainen 1989). These factors may have contributed to the finding by Niklas & Enquist (2001) that stand-average tree growth scaled isometrically with stand-average leaf biomass in a large data set across forest stands in different climates and site types.
Our attempt to maximize the metabolic exponent β in eqn 7, as suggested by West, Brown & Enquist (1999a), did not yield a simple prediction of the scaling exponent when the metabolic rate was constrained by self-shading. Instead, we find a more complex relationship between metabolic rate and body mass (eqn 7). It is possible to find the value for z that maximizes the scaling exponent β in eqn 7, but estimates of this optimal z depend strongly on other coefficients (in particular, k in eqn 3 and the coefficient of proportionality related to eqn 4). Previous work has considered the dynamic trade-off between crown size and self-shading, and the resulting optimal scaling of leaf area with body mass (Mäkelä & Sievänen 1992; Yamamura 1997). Consistent with our empirical results, these studies have found that the optimal fractal dimension is smaller than 3 but larger than 2, so that self-shading increases with body mass.
In summary, we have argued that the origin of allometric scaling in plants, especially trees, may be very different from that in animals owing to the differences in respective resource distribution and capture. We found in our theoretical analysis that the scaling of foliage mass or area with crown dimensions should have an intermediate exponent between area and volume scaling, a result also supported by our large data set of gymnosperm species. Despite this, we did not find large deviations in the actual scaling exponent d′ between foliage and total body mass (wood in crown, excluding the bole below the live crown) from the theoretical expectation of ¾ scaling, because we found compensatory deviations in scaling of woody crown biomass.
Results on foliage scaling increase our understanding of the dynamics of tree growth, and they have important applications in simulation models, for example, for projections of growth allocation (Valentine & Mäkelä 2005). They can also be utilized for estimating foliage mass, for example, for carbon stocks and fluxes (e.g. Lehtonen 2005). However, because we could not find a unique solution to the optimal allometric scaling problem, the optimal fractal dimension of the crown remains a challenge for future research. We suggest that the continuing efforts on finding mechanistic origins for scaling exponents in plants may utilize more dynamic optimization methods (Mäkelä & Sievänen 1992; Yamamura 1997) and combine multiple trade-offs between light-harvesting efficiency of foliage (Sterck & Schieving 2007), hydraulic supply to foliage (McCulloh & Sperry 2005), mechanical safety (West, Brown & Enquist 1999b) and the scaling of respiration and productivity with nutrient concentration (Niklas 2006; Reich et al. 2006) to arrive at a more complete picture of plant functioning in complex environments.
The authors thank Doug Maguire, Bob Monserud, John Marshall, Petteri Vanninen, Arne Albrektson, Sean Garber, Duncan Wilson and Boris Zeide for sharing their published data with them. Mikko Peltoniemi is acknowledged for providing useful comments on this manuscript. The Academy of Finland is gratefully acknowledged for funding the Meregrowth project (grant no. 106200).