The Horton-Strahler Number of Conditioned Galton-Watson Trees

TL;DR

This paper analyzes the growth of the Horton-Strahler number in conditioned Galton-Watson trees, revealing it grows logarithmically with tree size, and introduces related generalized measures with similar asymptotic behavior.

Contribution

It establishes the asymptotic growth of the Horton-Strahler number in critical Galton-Watson trees and introduces generalized versions with comparable properties.

Findings

Horton-Strahler number grows as (1/2) log_2 n in conditioned critical Galton-Watson trees.

Generalizations like the rigid Horton-Strahler number also exhibit similar asymptotic growth.

Provides theoretical results on the complexity measures of random trees.

Abstract

The Horton-Strahler number of a tree is a measure of its branching complexity; it is also known in the literature as the register function. We show that for critical Galton-Watson trees with finite variance conditioned to be of size $n$, the Horton-Strahler number grows as $\frac{1}{2}\log_2 n$ in probability. We further define some generalizations of this number. Among these are the rigid Horton-Strahler number and the $k$-ary register function, for which we prove asymptotic results analogous to the standard case.