Lengths of paths in rooted trees

Abstract

We provide formulas for generating functions of many types of paths in various rooted tree structures. We compute the $k$th moment of the generating functions for various types of vertical paths. In two specific familes of trees we find exact closed formulas for expectations and their asymptotic values. Some of these closed formulas are surprisingly simple.

We generalize the formulas used to give the $k$th moment of the length of vertical paths. The corresponding theorem was split into a theorem and a corollary for clarity. The title was changed to reflect the fact that the formulas apply to some non-plane trees as well