A full characterization of evolutionary tree topologies

The topologies of evolutionary trees are shaped by the nature of the evolutionary process, but comparisons of trees from different processes are hindered by the challenge of completely describing tree topology. We present a full characterization of the topologies of rooted branching trees in a form that lends itself to natural tree comparisons. The resulting metric distinguishes trees from random models known to produce different tree topologies. It separates trees derived from tropical vs USA influenza A sequences, indicating that the different epidemiology of tropical and seasonal flu leaves strong signatures in the tree topology. Our approach allows us to construct addition and multiplication on trees, and to create a convex metric on tree topologies which formally allows computation of average trees.