Radial heat diffusion from the root of a semi-homogeneous tree

We compute recursively the heat semigroup in a rooted homogeneous tree for the diffusion with radial (with respect to the root) but non-isotropic transition probabilities. This is the discrete analogue of the heat operator on the disc given by $\Delta + c \frac\partial{\partial r}$ for some constant $c$ that represents a drift towards (or away from) the origin.