A scaling law for the effects of architecture and allometry on tree vibration modes suggests a biological tuning to modal compartmentalization

Wind is a major ecological factor for plants and a major economical factor for forestry. Mechanical analyses have revealed that the multimodal dynamic behavior of trees is central to wind - tree interactions. Moreover, the trunk and branches influence dynamic modes, both in frequency and location. Because of the complexity of tree architecture, finite element models (FEMs) have been used to analyze such dynamics. However, these models require detailed geometric and architectural data and are tree-specific - two major restraints for their use in most ecological or biological studies. In this work, closed-form scaling laws for modal characteristics were derived from the dimensional analysis of idealized fractal trees that sketched the major architectural and allometrical regularities of real trees. These scaling laws were compared to three-dimensional FEM modal analyses of two completely digitized trees with maximal architectural contrast. Despite their simplifying hypotheses, the models explained most of the spatiotemporal characteristics of modes that involved the trunk and branches, especially for sympodial trees. These scaling laws reduce the tree to (1) a fundamental frequency and (2) one architectural and three biometrical parameters. They also give quantitative insights into the possible biological control of wind excitability of trees through architecture and allometries.