$k$-protected vertices in unlabeled rooted plane trees

TL;DR

This paper derives a closed-form formula for the proportion of $k$-protected vertices in unlabeled rooted plane trees and analyzes the asymptotic average ranks of vertices and roots.

Contribution

It provides a new simple formula for $k$-protected vertices and asymptotic average rank values in large unlabeled rooted plane trees.

Findings

Closed-form formula for $k$-protected vertices proportion

Asymptotic average vertex rank approaches 0.727649

Asymptotic root rank approaches 1.62297

Abstract

We find a simple, closed formula for the proportion of vertices which are $k$-protected in all unlabeled rooted plane trees on $n$ vertices. We also find that, as $n$ goes to infinity, the average rank of a random vertex in a tree of size $n$ approaches 0.727649, and the average rank of the root of a tree of size $n$ approaches 1.62297.