The path spectrum of a graph is the set of lengths of all maximal paths in the graph. A set 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 at least of all subsets of the set are spectral and conjecture that all the subsets with at least integers are spectral.