Consider an undirected graph G=(V,E) with n (=|V|) vertices and m (=|E|) edges. It is well-known that the problem of computing the sequence Nn-1,Nn,...,Nm is #P-complete (see e.g.,[3]), where Ni denotes the number of connected spanning subgraphs with i (n-1!im) edges in G. In this paper, by proving new inequalities on the sequence Nn-1,Nn,...,Nm, we show an interesting and stronger property that the sequence γn-1,γn,...,γm, where γi denotes the average number of spanning trees in the connected spanning subgraphs with i edges, is a convex sequence as well as a monotonically increasing sequence, although this property does not hold for the sequence Nn-1,Nn,...,Nm.