Path spectra for trees

The path spectrum   of a graph is the set of lengths of all maximal paths in the graph. A set SS of positive integers is spectral   if it is the path spectrum of a tree. We characterize the spectral sets containing at most two odd integers (and arbitrarily many even ones) and obtain several necessary conditions for a set to be spectral. We show that for each even integer s≥2s≥2 at least 1/41/4 of all subsets of the set {2,3,…,s}{2,3,…,s} are spectral and conjecture that all the subsets with at least 3s/43s/4 integers are spectral.