COMPUTER SIMULATION OF BRANCH INTERACTION AND REGULATION BY UNEQUAL FLOW RATES IN BOTANICAL TREES

Two aspects of branch interaction in trees are investigated theoretically. In the first it is assumed that there is a controlling factor in which the proximity of neighboring terminal branch units influences their branching capability. The almost horizontal tiers of lateral branches of Terminalia catappa L. and Cornus alternifolia L. are simulated by computer using values based on the measured branch geometries of real trees. For branch interaction, we assume a horizontal circle of inhibition whose center is the existing terminal point of a branch. If the end point of another branch extends into the circle, the original branch fails to bifurcate. Examples of computer simulated patterns are illustrated using different degrees of interference and are compared with branch tiers in T. catappa. In the second model the ability of a terminal branch unit to bifurcate is considered to be determined by the accumulation of a critical amount of a hypothetical growthor bifurcation-determining factor. The daughter branches of a bifurcation are assumed to have differing "flow rates,' i.e., the factor is distributed in different amounts between different daughter axes. Some simulated patterns generated by this model are very similar to real patterns found in T. catappa and an unnamed species of Tabernaemontana. In both simulations bifurcation ratios are determined and are shown to be a variable, not a fixed, property of the simulated trees. A TREE can be regarded conceptually as a system of axes which develops by a process of repeated branching or bifurcation. In botanical trees the frequency of branching is almost never the same at every terminal branch. Thus, the number of terminal branches does not increase by a regular exponent (2N) since the rate of increment normally decreases. In the simplest analysis, this decrease can be attributed to two theoretical constraints on growth: 1) an environmental or exogenous factor, resulting in inter-branch (inter-meristem) competition or interaction such that the actual number of permanent branches falls with increasing age as a result of abortion and abscission of both young and old branches, and 2) an intrinsic or endogenous factor, by which two daughter 1 Received for publication 18 April 1980; revision accepted 2 August 1980. The main part of the present work was carried out during the stay of H. Honda at Harvard Forest as a research fellow of the Charles Bullard Fund of Harvard University. Additional support was provided by the Atkins Garden Fund and Cabot Foundation of Harvard University. The contribution of J. B. Fisher was supported in part by NSF grants DEB77-13953 and 79-14635. Acknowledgement is due to Mmes. D. R. Smith (Harvard Forest), M. HayashiOkuda, and T. Kondo, Misses K. Fukuda, M. Noda, K. Tanaka, Y. Tanaka, and Y. Ueura (Kanebo Institute for Cancer Research) for assistance. 2 Permanent address: Kanebo Institute for Cancer Research, Misakicho 1-91, Kobe 652, Japan. branches from one mother branch differentiate into different kinds of branches, one of which bifurcates less frequently than the other. This fundamental difference in frequency of branching is explained by us theoretically as a result of different flow rates through branch pairs. In this paper we will investigate both constraints by using a tree model for computer simulation. The parameter values used in the model are obtained from three species of real trees, Cornus alternifolia, Tabernaemontana sp. and Terminalia catappa. Results of computer simulations are analyzed in terms of the number of terminal branches and the value of bifurcation ratios to allow some comparison with real trees. These ratios are indices which have been widely applied to tree systems, but we will show they have limited usefulness for understanding botanical trees. Computer simulation has shown that quite simple algorithms may be used to generate three-dimensional branching patterns which closely resemble actual trees (Honda, 1971). The method, therefore, offers techniques for investigating developmental rules that govern the form of botanical trees and, by varying chosen parameters one at a time, gives insight into the adaptive nature of tree form. The influence which variations in normal developmental processes can have on tree form can thus be investigated in a preliminary but very This content downloaded from 157.55.39.195 on Tue, 26 Apr 2016 05:54:56 UTC All use subject to http://about.jstor.org/terms 570 AMERICAN JOURNAL OF BOTANY [Vol. 68