Reversibility of a simple random walk on periodic trees

We show that if $\mu$ is a reversible measure for simple random walk on rooted trees whose branches are covers of finite connected directed graphs, then $\mu$ 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 $T$.