On the Steiner distance of trees from certain families

The Steiner distance dc(S) of a subset S of nodes of a connected graph G is the minimum number of edges in a connected subgraph of G that contains S. We consider the behaviour of the expected value ILk (n) of dT(S) and the probability Pk(n, m) that dT(S) rn lover all subsets S of k nodes of trees T of order n from certain families of trees.