On the profile of trees with a given degree sequence

TL;DR

This paper studies the profile of random rooted plane trees with a fixed degree sequence, providing conditions for their convergence and extending previous results through probabilistic methods and novel path transformations.

Contribution

It offers a general probabilistic framework for the profile of trees with given degree sequences, extending prior work and proving a conjecture by Aldous (1991).

Findings

Convergence conditions for the profile of trees with fixed degree sequences.

Extension of Aldous's conjecture on tree profiles.

A boundedness criterion for inhomogeneous continuum random trees.

Abstract

A degree sequence is a sequence ${\bf s}=(N_i,i\geq 0)$ of non-negative integers satisfying $1+\sum_i iN_i=\sum_i N_i<\infty$. We are interested in the uniform distribution $\mathbb{P}_{{\bf s}}$ on rooted plane trees whose degree sequence equals ${\bf s}$, giving conditions for the convergence of the profile (sequence of generation sizes) as the size of the tree goes to infinity. This provides a more general formulation and a probabilistic proof of a conjecture due to Aldous (1991). Our formulation contains and extends results in this direction obtained previously by Drmota and Gittenberger (1997) and Kersting (2011). A technical result is needed to ensure that trees with law $\mathbb{P}_{{\bf s}}$ have enough individuals in the first generations, and this is handled through novel path transformations and fluctuation theory of exchangeable increment processes. As a consequence, we obtain a boundedness criterion for the inhomogeneous continuum random tree introduced by Aldous, Miermont and Pitman (2004).