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Annals of Botany 73: 91-98, 1994
© 1994 Annals of Botany Company

A Simple Mathematical Model of Growth Pattern in Tree Stems

Yukihiro Chiba and Kichiro Shinozaki

Forestry and Forest Products Research Institute, Tsukuba, Ibaraki, 305 and Tezukayama University, Nara, 630 Japan

Stem formation process is analysed by examining annual rings for coniferous and broadleaved trees. Cutting a tree stem into several segments of a constant length, the weight of each segment is denoted as the stem density (S). In addition the vertically accumulated current increment of the stem densities is defined as the cumulative stem increment (CSI). Examining the relationships between CSI and S for tree stems, it is shown that most of them depict straight lines. Redefining S as a function with two independent variables, time t and vertical position z along the stem, the linear partial differential equation {partial}S/{partial}t = {kappa} {partial}S/{partial}z can be induced from the linear relations. This equation means that stem formation is founded substantially on a symmetrical pattern in stem increments along the direction of height growth and those along the course of time. The general solution of this equation implies that the tree stem should move upward at an arbitrary velocity {kappa} without changing its shape at all. In addition, a particular solution of the equation can be expressed by an exponential function that corresponds to a generalized form of the Oohata-Shinozaki model of stem shape. Analytical examinations of the linearity for the CSI-S relationships are made for testing them employing the data of stem segments. Reconsidering CSI for the past few years and combining stem cutting lengths with respect to height growth of the tree, it is clarified that the linearity for the CSI-S relationship be realized depending on their combinations.Copyright 1994, 1999 Academic Press

Cumulative stem increment, height growth, ring analysis, stem density, stem growth equation, symmetric growth pattern


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