Transience and volumes of trees (Q1061420)

Let T be a locally finite, countably infinite tree with no vertex of degree one. A relationship between probabilistic and geometric properties of T is exposed: It is proved that for the simple random walk on T, the mean number of return visits to a given starting vertex is the minimal value among all generalized volumes of T and is in particular less or equal to the ordinary volume of the tree. A ''geometrical'' criterion for transience follows.