Path spectra for trees

The path spectrum of a graph is the set of lengths of all maximal paths in the graph. A set S 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 ≥ 2 at least 1 / 4 of all subsets of the set { 2 , 3 , … , s } are spectral and conjecture that all the subsets with at least 3 s / 4 integers are spectral.