Limits of biconditioned Bienayme-Galton-Watson trees

TL;DR

This paper investigates the asymptotic structures of Bienayme-Galton-Watson trees under specific conditioning, revealing universal and diverse behaviors depending on the constraints and offspring distribution.

Contribution

It provides a comprehensive analysis of the limiting behaviors of biconditioned Bienayme-Galton-Watson trees, highlighting universal and varied phenomena.

Findings

Universal limiting behavior for trees conditioned on fixed number of leaves.

Diverse limiting structures, including condensation, depending on offspring distribution.

Application of conditioned random walk and analytic combinatorics tools.

Abstract

We study the limiting behavior of a Bienayme-Galton-Watson tree conditioned to have a large number of vertices and either a fixed number of leaves or a fixed number of internal nodes. The first biconditioning gives a universal result with respect to the offspring distribution. In contrast, the second case leads to a variety of limiting behaviors, ranging from condensation phenomena to more elongated tree structures, depending on the properties of the offspring distribution. To prove these results, we use tools from conditioned random walk theory and from analytic combinatorics.