A congruence theorem for trees.

Let A and JB be two trees with vertex sets au α2, , an and bi, b2, • ••, bn respectively. The trees are congurent, are isomorphic, or "are the same type", (Aζ^B), if there exists a one-to-one correspondence between their vertices which preserves the join-relationship between pairs of vertices. Let c(at) denote the (n-l)-point subgraph of A^obtained by deleting at and all joins (arcs, segments) at at from A. It is the purpose here to show that if there is a one-to-one correspondence in type, and frequency of type, between the sub-graphs of order n — lmA and J5, that is, if there exists a labeling such that φ ^ ) ^ cφi), i = l , 2, •••, n, then A ~ B. It is assumed throughout, therefore, that there is a labeling of the two trees A and B such that c(α4)^c(δ4), i = l , 2, •••, n9 where n^Z. Some lemmas to the main theorem are established first. Let T denote a certain type of graph of order j , where 2<Lj <^n, which occurs as a subgraph a times in A and β times in B. If at is the number of T-type subgraphs which have at as a vertex, then,