Homometric sets in trees

Let G=(V,E) denote a simple graph with vertex set V and edge set E. The profile of a vertex set V′⊆V denotes the multiset of pairwise distances between the vertices of V′. Two disjoint subsets of V are homometric if their profiles are the same. If G is a tree on n vertices, we prove that its vertex set contains a pair of disjoint homometric subsets of size at least n/2−1. Previously it was known that such a pair of size at least roughly n1/3 exists. We get a better result in the case of haircomb trees, in which we are able to find a pair of disjoint homometric sets of size at least cn2/3 for a constant c>0.