Asymptotics of trees with a prescribed degree sequence and applications

Let t be a rooted tree and n_bi(t) the number of nodes in t having i children. The degree sequence of t satisfies , where denotes the number of nodes in t. In this paper, we consider trees sampled uniformly among all plane trees having the same degree sequence ; we write for the corresponding distribution. Let be a list of degree sequences indexed by κ corresponding to trees with size . We show that under some simple and natural hypotheses on the trees sampled under converge to the Brownian continuum random tree after normalisation by . Some applications concerning Galton–Watson trees and coalescence processes are provided.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 290‐316, 2014