Minimum graphs that contain all small trees.

Let E sub n denote the minimum number of edges in a graph that contains every tree with n edges. A new lower bound on E sub n, defined by a dynamic optimization procedure is computed for all n = 61 and is shown to equal E sub n at least through n = 11. Two further things are determined for graphs on n + 1 vertices with E sub n edges for each n = 9: (a) a minimum set of trees with n edges such that all trees with n edges are contained in the graph whenever it contains the trees in the minimum set; (b) all mutually nonisomorphic graphs that contain all trees with n edges.