Asymptotic of trees with a prescribed degree sequence and applications

• Published in: Random structures & algorithms Vol. 44; no. 3
• Main Authors: Broutin, Nicolas, Marckert, Jean-François
• Format: Journal Article
• Language: English
• Published: Wiley 2014
• Series: Random Structures and Algorithms
• Subjects: Mathematics; Probability
• ISSN: 1042-9832; 1098-2418

Let t be a rooted tree and ni(t) the number of nodes in t having i children. The degree sequence (ni(t),i≥0) of t satisfies ∑i≥0ni(t)=1+∑i≥0ini(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 κ 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.