Asymptotics of trees with a prescribed degree sequence and applications

TL;DR

This paper investigates the asymptotic behavior of large trees with fixed degree sequences, demonstrating convergence to the Brownian continuum random tree under certain conditions, with applications to Galton-Watson trees and coalescence processes.

Contribution

It establishes conditions under which trees with prescribed degree sequences converge to the Brownian continuum random tree as size grows.

Findings

Trees with fixed degree sequences converge to the Brownian CRT after normalization.

Results apply to Galton-Watson trees and coalescence processes.

Provides a framework for understanding large random trees with degree constraints.

Abstract

Let $t$ be a rooted tree and $n_i(t)$ the number of nodes in $t$ having $i$ children. The degree sequence $(n_i(t),i\geq 0)$ of $t$ satisfies $\sum_{i\ge 0} n_i(t)=1+\sum_{i\ge 0} in_i(t)=|t|$, where $|t|$ denotes the number of nodes in $t$. In this paper, we consider trees sampled uniformly among all trees having the same degree sequence $\ds$; we write $`P_\ds$ for the corresponding distribution. Let $\ds(\kappa)=(n_i(\kappa),i\geq 0)$ be a list of degree sequences indexed by $\kappa$ corresponding to trees with size $\nk\to+\infty$. We show that under some simple and natural hypotheses on $(\ds(\kappa),\kappa>0)$ the trees sampled under $`P_{\ds(\kappa)}$ converge to the Brownian continuum random tree after normalisation by $\nk^{1/2}$. Some applications concerning Galton--Watson trees and coalescence processes are provided.