Every tree T determines a set of distinct maximal proper subtrees Ti = T — vi , which are obtained by the deletion of an endpoint of T. In this paper we prove that a tree is almost always uniquely determined by this set of its subtrees, and point out two interesting consequences of this result.
In , Ulam proposed the following conjecture, which we state in a slightly stronger form due to Harary .
ULAM'S CONJECTURE. A graph G with at least three points is uniquely determined up to isomorphism by the subgraphs Gi = G — vi .
Kelly  proved the conjecture for trees and Harary and Palmer  showed that not all of the Gi are needed in that case by proving Corollary 1 below. If we remove from the list of subgraphs Gi of a graph G all but one graph of each isomorphism type, we obtain a set of Gi which are distinct up to isomorphism.