Asymptotics of trees with a prescribed degree sequence and applications

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Abstract

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

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