The shapes of evolutionary trees are influenced by the nature of the evolutionary process, but comparisons of trees from different processes are hindered by the challenge of completely describing tree shape. We present a full characterization of the shapes 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 shapes. It separates trees derived from tropical vs USA influenza A sequences, which reflect the differing epidemiology of tropical and seasonal flu. We extend the shape metric to incorporate summary features such as asymmetry, or statistics on branch lengths. Our approach allows us to construct addition and multiplication on trees, and to create a convex metric on tree shapes which formally allows computation of average trees.