### Tree Shape and Branch Structure: Mathematical Models

Dick F Yeatts

#### Abstract

This study of tree morphology is presented in three parts. Part 1 deals with the over-all shape of trees (their profiles). A mathematical model based on the distribution of leaves shows that all profiles, from rounded to conical, is determined by one parameter. Part 2 deals with the angle of branching. Field measurements show that the sum of the areas of stems exiting a fork is usually greater than the area of the stem entering the fork. Mathematical analysis shows that this â€œbulking upâ€ actually reduces the quantity of plant tissue incorporated in the branching. Furthermore, the angle of branching increases with the degree of bulking up. Part 3 brings together the concepts of the first two parts: here, the cross-sectional area of the trunk as a function of position along the trunk is determined by the profile of the tree and the angle of side branching off the trunk. From field observation, the profile of a tree (i.e., the outline of its crown) has two prominent characteristics: (i) azimuthal symmetry about the central axis (often the main stem or trunk), as evidenced both in foliage and scaffolding; and (ii) decrease in leaf density from branch-end toward the central axis. From these two conditions (including a presumed radial distribution of leaves) a mathematical model is developed using the calculus of variations that predicts the profile The results of this analysis are consistent with the general observation that 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. Again, using the calculus of variations, it is shown that the angle of branching can be related to this â€œbulking upâ€. Field measurements are in general agreement with this prediction. Part 3 provides a separate test of the validity of the concepts introduced in the first two Parts. Using equations appropriate to a tree with a single main stem and horizontal side branches, the crosssectional area of the trunk is calculated as a function of position on the trunk. The results are compared with field measurements.

#### Keywords

optimization, modeling, tree shape

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