Increasing paths in regular trees

TL;DR

This paper studies the probability of increasing label paths in regular trees, showing that for fixed ratio n/h > 1/e, such paths almost surely exist as the tree height grows, with implications for evolutionary biology.

Contribution

It establishes a phase transition in the existence of increasing paths in regular trees based on the ratio n/h, extending previous results.

Findings

Probability of increasing paths approaches 1 if n/h > 1/e as height increases.

Probability approaches 0 if n/h ≤ 1/e as height increases.

Identifies a critical threshold for path existence in regular trees.

Abstract

We consider a regular $n$-ary tree of height $h$, for which every vertex except the root is labelled with an independent and identically distributed continuous random variable. Taking motivation from a question in evolutionary biology, we consider the number of simple paths from the root to a leaf along vertices with increasing labels. We show that if $\alpha = n/h$ is fixed and $\alpha > 1/e$, the probability there exists such a path converges to 1 as $h \to \infty$. This complements a previously known result that the probability converges to 0 if $\alpha \leq 1/e$.