The Horton-Strahler number of Galton-Watson trees with possibly infinite variance

TL;DR

This paper analyzes the growth and behavior of the Horton-Strahler number in critical Galton-Watson trees with offspring distributions in the domain of attraction of stable laws, extending previous finite variance results to infinite variance cases.

Contribution

It provides new probabilistic proofs characterizing the Horton-Strahler number for trees with infinite variance offspring distributions, including the critical case where quals 1.

Findings

Horton-Strahler number grows logarithmically with tree size for 

Provides tail estimates for the Horton-Strahler number in these trees

Characterizes complex behaviors in the  Cauchy regime

Abstract

The Horton-Strahler number, also known as the register function, provides a tool for quantifying the branching complexity of a rooted tree. We consider the Horton-Strahler number of critical Galton-Watson trees conditioned to have size $n$ and whose offspring distribution is in the domain of attraction of an $\alpha$-stable law with $\alpha\in [1, 2]$. We give tail estimates and when $\alpha\neq 1$, we prove that it grows as $\frac{1}{\alpha}\log_{\alpha/(\alpha-1)} n$ in probability. This extends the result in Brandenberger, Devroye \& Reddad [6] dealing with the finite variance case for which $\alpha=2$. We also characterize the cases where $\alpha=1$, namely the spectrally positive Cauchy regime, which exhibits more complex behaviors. Our proofs are new and probabilistic; they relate the Horton-Strahler number with other shape parameters such as the height or largest degree.