= 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> give
$$
w_-^{\rm rad}=\frac1z,\qquad w_+^{\rm rad}=\frac{z-1}{z}.
$$
Let $r$ 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:
$$
\boxed{P(r,N+1)=\frac{z-1}{z}P(r-1,N)+\frac1zP(r+1,N),\qquad r>1.}
$$
This has precisely the same bulk <master equation> as the one-dimensional <biased random walk>, with
$$
\boxed{\epsilon=\frac{w_-^{\rm rad}-w_+^{\rm rad}}2=\frac{2-z}{2z}.}
$$
The 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.
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