Lower Bounds on the Odds Against Tree Spectral Sets

The path spectrum of a graph is the set of lengths of all maximal paths in the graph. A set S of positive lengths is tree spectral if it is the path spectrum of a tree. We show that for each even integer s⩾2 at least 34.57% of all subsets of the set {2,3,…,s} are tree spectral, and for each odd integer s⩾2 at least 27.44% of all subsets of the set {2,3,…,s} are tree spectral.