Central Limit Theorem of the Bifurcation Ratio of the Horton–Strahler Analysis: Application to DLA Clusters

Abstract

The method of branch ordering evaluates the hierarchical structure of branching patterns, and the bifurcation ratio is an important variable in the topological analysis of branching patterns. In a simple mathematical model of a branching system called the random model, a central limit theorem for the inverse bifurcation ratio has been confirmed. In the present paper, we make a qualitative comparison between the random model and an ensemble of diffusion-limited aggregation (DLA) clusters from the viewpoint of the central limit theorem. From numerical data, we extrapolate asymptotic behaviors of the average and variance of the inverse bifurcation ratio of DLA clusters. These asymptotic properties are in good agreement with those of the random model qualitatively. Hence, the central limit theorem also holds for DLA clusters.