Radial random walk on a regular tree ($p_{\mathrm{out}}=(z-1)/z,p_{\mathrm{in}}=1/z$)

For a simple random walk on an infinite regular tree of degree $z\ge 2$, every vertex other than the chosen root has one neighbour closer to the root and $z-1$ farther away. The graph distance from the root is therefore a nearest-neighbour Markov chain on the nonnegative integers, with the displayed bulk transition probabilities and a certain outward step from zero. For shell probabilities $P_r(N)$,Shell probabilities include all vertices at the given distance. They differ from single-vertex probabilities by the shell size, $1$ at the root and $z(z-1{)}^{r-1}$ for $r\ge 1$.