Embedding frequencies of trees

A graph is said to be embedded in a graph if is isomorphic to a subgraph of . The embedding frequency for in ,N( , ), is the number of different subgraphs of to which is isomorphic. We use a computer program to calculate the embedding frequencies of subtrees within trees. We computeN(, ) for trees through 10 vertices and present the results in tabular form. When trees are partially ordered by valence class, their subtrees lie in corresponding order; we give a formal proof of this subtree embedding property. The structure of the embedding relation is exhibited in a topological picture of the zeta function showing the non-zero values ofN(, ).