Distances between pairs of vertices and vertical profile in conditioned Galton--Watson trees

TL;DR

This paper analyzes the structure of conditioned Galton-Watson trees, providing estimates for vertex pair distances and proving the convergence of the vertical profile to ISE density, using probabilistic and analytical methods.

Contribution

It introduces two proofs for counting vertex pairs at given distances and confirms a conjecture about the vertical profile's distribution convergence in conditioned Galton-Watson trees.

Findings

Estimate of the number of vertex pairs at given distances

Proof of convergence of the vertical profile to ISE density

Two different proof techniques for the main results

Abstract

We consider a conditioned Galton-Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis. Moreover, the second proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet-Melou and Janson saying that the vertical profile of a randomly labelled conditioned Galton-Watson tree converges in distribution, after suitable normalization, to the density of ISE (Integrated Superbrownian Excursion).