K-Protected Vertices in Unlabeled Rooted Plane Trees

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.