The distance to the border of a random tree

TL;DR

This work characterizes the distance-to-border parameter $\partial(T)$ for conditioned Galton–Watson trees by deriving a precise asymptotic formula for $\mathbf{P}(\partial(T)\ge k\mid \#(T)=n)$ as $n\to\infty$, and relates it to the height via a unifying Lagrange-inversion framework. The authors develop an iterative scheme based on Lagrange’s equation within the theory of Khinchin families, enabling explicit asymptotics for the coefficients of iterates and enabling transfer to various tree families through analytic combinatorics. The results yield computable constants for classical models (Cayley, plane, binary plane trees) and provide mean-structure consequences, such as the asymptotic proportion and expected counts of nodes whose distance-to-border exceeds $k$. The approach links the theory of Khinchin families with simply generated trees, offering a versatile method to analyze local structural parameters in random trees with potential broad applicability to related combinatorial-probabilistic models.

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 Rényi 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.

Figures (1)

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Theorems & Definitions (39)

• Theorem A: Meir-Moon
• Corollary A
• Lemma 3.1
• proof
• Lemma A
• Lemma 3.2
• proof
• Remark 3.3
• Theorem 4.1
• Remark 4.2