Geodesic
Radial Heat Diffusion from the Root of a Homogeneous Tree and the Combinatorics of Paths
Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 619-628.

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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. 
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Cohen, Joel M.; Pagliacci, Mauro; Picardello, Massimo A. Radial Heat Diffusion from the Root of a Homogeneous Tree and the Combinatorics of Paths. Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 619-628. http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a5/

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