Crown Ratio Influences Allometric Scaling in Trees

Abstract

Allometric theories suggest that the size and shape of organisms follow universal rules, with a tendency toward quarter-power scaling. In woody plants, however, structure is influenced by branch death and shedding, which leads to decreasing crown ratios, accumulation of heartwood, and stem and branch tapering. This paper examines the impacts on allometric scaling of these aspects, which so far have been largely ignored in the scaling theory. Tree structure is described in terms of active and disused pipes arranged as an infinite branching network in the crown, and as a tapering bundle of pipes below the crown. Importantly, crown ratio is allowed to vary independently of crown size, the size of the trunk relative to the crown deriving from empirical results that relate crown base diameter to breast height diameter through crown ratio. The model implies a scaling relationship in the crown which reduces to quarter-power scaling under restrictive assumptions but would generally yield a scaling exponent somewhat less than three-quarters. For the whole tree, the model predicts that scaling between woody mass and foliage depends on crown ratio. Measurements on three boreal tree species are consistent with the model predictions.

Tree or shrub: a functional branch analysis of Jatropha curcas L.

“…Biologically, this relationship is complex as link length is determined by the development and disappearance (self-pruning) of branches, in the axes of (former) leaves. Enhanced development of side branches at the crown surface leads to shortening of link lengths, while shading of the lower branches leads to senescence of side branches (Mäkelä and Valentine 2006) and secondary elongation of links as measured in the FBA protocol. Such trees can be thought of as 'disrupted fractals' where the lengths of links of different diameter branches are somewhat randomised due to historical branch development that effects link length (MacFarlane et al 2014).…”

“…Fractal (self-similarity) properties of the branches are restricted to the crown (Mäkelä 2002;Mäkelä and Valentine 2006). Similar to FBA for forest tree, jatropha biomass was constructed in the same way, with addition of fruit biomass.…”

Precision of allometric scaling equations for trees can be improved by including the effect of ecological interactions

“…This variability can be accounted for by measuring additional ecological factors at the level of the individual. Fitting both exponents and normalizations at the level of the individual (Mäkelä and Valentine 2006;Price et al 2007), and accounting for relationships between these parameters and ecological factors, could even further increase the predictive power of allometries. However, because of the strong interdependence between the two parameter values (Lumer 1939), one of the scaling parameters may have to be set to a predetermined value before the value of the other can be accurately estimated using merely statistical fit.…”

“…There have been only a few attempts to give a mechanistic explanation for the statistical variability although both the scaling exponent and the normalization constant may have an interpretation based on biological processes (Kozłowski et al 2003;Etienne et al 2006;Mäkelä and Valentine 2006;Chown et al 2007;Enquist et al 2007;Price et al 2007). As an alternative, some process-based models use conditional values of a and b, or additional variables to modify the values of a and b, instead of attempting to give a direct interpretation of the values of a and b by themselves (Duursma et al 2007;Holdo 2007).…”

“…For example, the scaling between woody mass and foliage mass seems to be strongly linked to the crown ratio of trees (Mäkelä and Valentine 2006). Crown ratio together with other crown parameters are, in turn, influenced by the amount of competition in the neighbourhood (Ilomäki et al 2003;Kantola and Mäkelä 2004).…”