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.