Tree Shape and Branch Structure: Mathematical Models

This study of tree morphology is presented in three parts: Part 1 deals with the profile (or envelope) of trees and woody plants. Noting that most trees exhibit: (i) azimuthal symmetry about the central axis (often the main stem or bole) in both in foliage and scaffolding; and (ii) decrease in leaf density from branch-end toward the central axis, a mathematical model is developed using the calculus of variations that predicts the profile, with but one free parameter. The analysis predicts profiles range from the nearly spherical in the case of uniform distribution of leaves throughout the crown, to essentially conical when the leaves are found largely on the branch-ends. The results are presented in figures showing theoretical profiles overlaid on photographs of representative trees. Part 2 is based on field measurements that show that the cross-sectional area of a branch (or stem) entering a fork (in the direction of water transport) is less than the sum of the cross-sectional areas of the branches leaving that fork. Mathematical analysis using the calculus of variations shows that this “bulking up” actually reduces the quantity of plant tissue incorporated in the branching. Furthermore, it is shown that the angle of branching increases with bulking up. Field measurements are in rough agreement with this prediction. Part 3 brings together the concepts of the first two parts to predict the cross-sectional area of the bole as a function of longitudinal position. Using equations appropriate to a tree with a single main stem and horizontal side branches, the cross-sectional area of the bole is calculated. The results compare favorably with field measurements.