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In the infinite regular tree $\mathbb{T}_{q+1}$ with $q \in \mathbb{Z}_{\ge2}$, we consider families $\{\mu_u^n\}$, indexed by vertices $u$ andnonnegative integers ("discrete time steps") $n$, of probability measures suchthat $\mu_u^n(v) = \mu_{u'}^n(v')$ if the distances $\operatorname{dist}(u,v)$and $\operatorname{dist}(u',v')$ are equal. Let $d$ be a positive integer, andlet $X$ and $Y$ be two vertices in the tree which are at distance $d$ apart. Wecompute a formula for the transportation distance $W_1\!\left( \mu_X^n, \mu_Y^n\right)$ in terms of generating functions. In the special case where $\mu_u^n =\mathfrak{m}_u^n$ are measures from simple random walks after $n$ time steps,we establish the linear asymptotic formula $W_1\!\left( \mathfrak{m}_X^n,\mathfrak{m}_Y^n \right) = An + B + o(1)$, as $n \to \infty$, and give theformulas for the coefficients $A$ and $B$ in closed forms. We also obtainlinear asymptotic formulas in the cases of spheres and uniform balls as theradii tend to infinity. We show that these six coefficients (two from simplerandom walks, two from spheres, and two from uniform balls) are related byinequalities.

Transportation Distance between Probability Measures on the Infinite Regular Tree