Power-law Estimation of Branch Growth

We demonstrate the efficacy of power-law models in the analysis of tree branch growth. The models can be interpreted as allometric equations, which incorporate multiple driving variables in a single scaling relationship to predict the amount of growth within a branch. We first used model selection criteria to identify the variables that most influenced (1) the length of individual elongating annual shoots and (2) the total length of all elongating annual shoots in the individual branches of silver birch (Betula pendula Roth). We then applied the two resulting power-law equations as dynamic models to predict the trajectories of crown profile development and accumulation of branch biomass during tree growth, using total branch length as a proxy for biomass. In spite of the wide size range and geographical distribution of the study trees, the models successfully reproduced the dynamic characteristics of crown development and branch biomass accumulation. Applying the model to predict long-term growth of a single branch that was initiated at the crown top generated a realistic crown profile and produced a final basal branch size that was well within the range of field observations. The models also predicted a set of more subtle and non-trivial features of crown formation, including the increased rate of growth towards the tree apex, decrease in growth towards the lowest branches, the effect of branching order on the amount of elongation, and the higher vigour of thick branches when the effect of branch height was controlled. In contrast, a simple allometric model of the form Y = aXb was incapable of capturing all the variability in growth of individual branches and of predicting the features of crown shape and branch size that are associated with the slowing-down of growth towards the crown base. We conclude that power-law models where the parameter a is refined to include spatial information on branch features shows good potential for identifying and incorporating actual crown construction processes in dynamic models that utilize the structural features of tree crowns.