The type problem for random walks on trees

We consider the question of whether the simple random walk (SRW) on an infinite tree is transient or recurrent. For random-trees (all vertices of distancen from the root of the tree have degreed_n, where {d_n} are independent random variables), we prove that the SRW is a.s. transient if lim inf_nnE(log(d_n-1))>1 and a.s. recurrent if lim sup_nn E(log(d_n-1))<1. For random trees in which the degrees of the vertices are independently 2 or 3, with distribution depending on the distance from the root, a partial classification of type is obtained.Research supported in part by NSF DMS 8710027.