The distance to the border of a random tree

TL;DR

This paper investigates the asymptotic probability that a conditioned Galton-Watson tree has a large distance to its border, extending classical results on tree height and leaf distances to a new probabilistic setting.

Contribution

It introduces the study of the distance to the border in conditioned Galton-Watson trees, providing asymptotic probability results and connecting to classical tree distance problems.

Findings

Asymptotic probability estimates for large distance to border

Extension of classical tree height results to border distance

Connection between border distance and leaf distance extremes

Abstract

Given a Galton-Watson process conditioned to have total progeny equal to $n$, we study the asymptotic probability that this conditioned Galton-Watson process has distance to the border bigger or equal than $k$, as the number of nodes $n \rightarrow \infty$. A problem which is akin to this one was solved by R\'enyi and Szekeres for Cayley trees, de Bruijn, Knuth, and Rice for plane trees and Flajolet, Gao, Odlyzko, and Richmond for binary trees. The distance to the border is dual, in a certain sense, to the height. The first of these distances is the minimum of the distances from the root to the leaves. The second is the maximum of the distances from the root to the leaves. These are two extreme complementary cases.