On the number of walks on a regular Cayley tree

TL;DR

This paper derives new generating functions for counting walks on regular Cayley trees, extending classical results and providing a fresh perspective on well-studied combinatorial structures.

Contribution

It offers a novel derivation of generating functions for walks on regular Cayley trees, connecting historical results with new analytical methods.

Findings

New derivation of generating functions for walks on regular trees

Extension of classical results to more general walk counts

Historical connection to Kesten and McKay's work

Abstract

We provide a new derivation of the well-known generating function counting the number of walks on a regular tree that start and end at the same vertex, and more generally, a generating function for the number of walks that end at a vertex a distance i from the start vertex. These formulas seem to be very old, and go back, in an equivalent form, at least to Harry Kesten's work on symmetric random walks on groups from 1959, and in the present form to Brendan McKay (1983).