Numerical Approach to Central Limit Theorem for Bifurcation Ratio of Random Binary Tree

TL;DR

This paper numerically investigates a central limit theorem for the bifurcation ratio in random binary trees, using binary sequences and Horton-Strahler indices to derive variances and confirm Gaussian distribution fitting.

Contribution

It introduces a numerical approach to analyze the CLT for binary trees, expressing their structure via sequences and deriving variance formulas.

Findings

Gaussian distribution fits numerical data well

Variances are expressed in simple formulas

Two types of CLT are formulated for higher-order branches

Abstract

A central limit theorem for binary tree is numerically examined. Two types of central limit theorem for higher-order branches are formulated. A topological structure of a binary tree is expressed by a binary sequence, and the Horton-Strahler indices are calculated by using the sequence. By fitting the Gaussian distribution function to our numerical data, the values of variances are determined and written in simple forms.