Subtrees of graphs

We study a new graph invariant, the sequence {sk} of the number of k‐edge subtrees of a graph. We compute the mean subtree size for several classes of graphs, concentrating on complete graphs, complete bipartite graphs, and theta graphs, in particular. We prove that the ratio of spanning trees to all subtrees in Kn approaches (1/e)(1/e)=0.692201⋯, and give a related formula for Kn,n. We also connect the number of subtrees of Kn that contain a given subtree to the hyperbinomial transform. For theta graphs, we find formulas for the mean subtree size (approximately 23n) and the mode (approximately 22n) of the unimodal sequence {sk}. The main tool is a subtree generating function.