Mathematical modelling to determine the greatest height of trees

Abstract

This study aimed to analyse the critical height of a column whose weight varies vertically in order to obtain a simple scaling law for a tree where the weight distribution considered. We modelled trees as cantilevers that were fixed to the ground and formulated a self-buckling problem for various weight distributions. A formula for calculating the critical height was derived in a simple form that did not include special functions. We obtained a theoretical clarification of the effect of the weight distribution of heavy columns on the buckling behaviour. A widely applicable scaling law for trees was obtained. We found that an actual tree manages to distribute the weight of its trunk and branches along its vertical extent in a manner that adequately secures its critical height. The method and findings of this study are applicable to a wide range of fields, such as the simplification of complicated buckling problems and the study of tree shape quantification.

Self-weight buckling of thin elastic shells: the case of a spherical equatorial segment

“…A simple case in point is a vertical cylindrical rod fixed at the lower end and free at the top; here one would be interested in the maximum length for which the rod maintains a straight vertical shape; alternatively, if the length is prescribed, another possible question of interest would boil down to identifying the minimum cross-sectional radius for which the previous property holds true as well. This problem was first solved by Greenhill [4] in connection to how tall a tree can grow (for related and more recent work see [5,6,7]).…”

Mechanics-based classification rule for plants

“…This formula indicates that the greatest height of a solid cylinder against self-weight buckling is proportional to two-thirds of the power of its diameter ( 1 ), and McMahon demonstrated that the scaling law of height–diameter based on Greenhill’s theory is applicable to actual trees as well ( 2 , 3 ). Through these verification results, Greenhill’s achievements have been widely applied in the fields of ecology, forest science, and engineering ( 5 – 24 ).…”

Rigidity control mechanism by turgor pressure in plants

“…Notwithstanding this diversity of form, there is a scaling law that is consistently present across various types of plants. This ubiquitous law may be ascribed to plants’ common goal of “efficiently resisting gravity” 1 . Greenhill derived a theoretical expression for the greatest height that a tree can grow to without buckling under its own weight, finding that this greatest height is proportional to the 2/3 power of the tree’s radius 2 .…”