We try to classify reversible measures for simple random walk on trees with branches that are multi-type Galton-Watson trees.

We show that if μ is a reversible measure for simple random walk on rooted trees whose branches are covers of finite connected directed graphs, then μ is supported on rooted covers of finite connected undirected graphs. For a given finite connected directed graph G and a cover T of G, we give an algorithm to determine whether there exists a finite connected undirected graph whose cover has a branch isomorphic to T.

Fix n ∈ [special characters omitted]. Let T_n be the set of rooted trees (T, o) whose vertices are labeled by elements of {1,…, n}. Let ν be a strongly connected multi-type Galton-Watson measure. We give necessary and sufficient conditions for the existence of a measure μ that is reversible for simple random walk on T _n and has the property that given the labels of the root and its neighbors, the descendant subtrees rooted at the neighbors of the root are independent multi-type Galton-Watson trees with conditional offspring distributions that are the same as the conditional offspring distributions of ν when the types of ν are ordered pairs of elements of [ n]. If the types of ν are given by the labels of vertices, then we give an explicit description of such μ.