  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).
