 Solution

For this radial random walk on a regular tree, the drawing is an infinite regular tree of degree $z$, not a planar lattice with closed loops. The radial coordinate is the number of edges in the unique path to the root. A non-root vertex has one neighbour inward and $z-1$ neighbours outward, so equal probabilities for its $z$ edges giveLet $r$ be radial distance measured in edge units. For $r>1$, probability arriving in shell $r$ comes from shell $r-1$ by an outward step or from shell $r+1$ by an inward step:This has precisely the same bulk master equation as the one-dimensional biased random walk, withThe outward bias is geometric: beyond the first shell, the number of vertices is $M_r=z(z-1{)}^{r-1}$, so there are more outward than inward choices even though no edge direction is dynamically preferred. $P(r,N)$ is the probability summed over an entire shell, not the probability at one specified vertex; no additional shell-size factor belongs in the boxed recurrence.

The equality concerns the evolution equation away from the root, not equality to the unrestricted full-line binomial distribution at all positions. The root imposes a different boundary rule.