  Null models of binary phylogenetic trees are useful for testing hypotheses on
real world phylogenies. In this paper we consider phylogenies as binary trees
without edge lengths together with a sampling measure and encode them as
algebraic measure trees. This allows to describe the degree of similarity
between actual and simulated phylogenies by focusing on the sample shape of
subtrees and their subtree masses. We describe the annealed law of the
statistics of subtree masses of null models, namely the branching tree, the
coalescent tree, and the comb tree in more detail. Finally, we use methods from
martingale problems to characterize evolving phylogenetic trees in the
diffusion limit.
