Topological self-similarity on the random binary-tree model

TL;DR

This paper analyzes the hierarchical structure of random binary trees using Horton-Strahler analysis, proving topological self-similarity asymptotically and introducing a recursive transformation of binary trees.

Contribution

It introduces a new recursive transformation of binary trees and proves topological self-similarity in the random binary-tree model.

Findings

Topological self-similarity is established asymptotically.

A recursive equation for branch orders is derived.

Examples illustrating the concepts are provided.

Abstract

Asymptotic analysis on some statistical properties of the random binary-tree model is developed. We quantify a hierarchical structure of branching patterns based on the Horton-Strahler analysis. We introduce a transformation of a binary tree, and derive a recursive equation about branch orders. As an application of the analysis, topological self-similarity and its generalization is proved in an asymptotic sense. Also, some important examples are presented.