Null models of binary genealogical or phylogenetic trees are useful for testing hypo- theses. In this chapter we describe the space of algebraic measure trees whose elements represent phylogenies and genealogies as binary trees without edge lengths endowed with a sampling measure. With the aim of describing the degree of similarity between actual and simulated phylogenies or genealogies, we focus on the sample shape of subtrees and related statistics. We describe certain statistics of the branching and coalescent tree in more detail. Finally, we use the martingale problem method to characterise evolving trees analytically.