I recently encountered this problem. Frankly I'm stuck; would be nice for some help. Here it is:
Let $N,k$ be positive integers. By $p_k(N)$ we denote the number of integer partitions of $N$ with exactly $k$ parts; that is, the number of ways to write $N$ as a sum of exactly $k$ positive integers, where the order of summands does not matter. What is the number of pairwise non-isomorphic spanning trees of $W_n$ of maximum degree at least 4 and with at most 1 vertex of degree 3? Write this number using terms of the form $p_k(N)$.