Let G be a connected graph, S a subset of its vertices . The Steiner distance d(S,G) of S in G is the minimum number of edges in any connected subgraph of G that contains S. The total Steiner k-distance D(k;G) (k=2,3,...) is the sum of d(S,G) over all subsets S of k vertices in G. (When k=2, the quantity is known as Wiener index of G). The purpose of this talk is to present asymptotic results (for large n): (i) For the expected value M(k;n) of d(S,T) over all randomly chosen subsets of k nodes in trees T with n nodes, which belong to certain (infinite) families F of random rooted trees. (ii) For p(k;n,m), the probability that d(S,T)=m-1, when |S|=k fixed, |T|=n and T belongs to a family F. (joint work with L. Clark and J.W. Moon)