Abstract In this article we study the theories of the infinite-branching tree and the r -regular tree, and show that both of them are pseudofinite. Moreover, we show that they can be realized by infinite ultraproducts of polynomial exact classes of graphs, and provide a characterization of the Morley rank of definable sets in terms of the degrees of polynomials measuring their non-standard cardinalities. This answers negatively some questions from [2], where it is asked whether every stable generalised measurable structure is one-based.

Abstract Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different notions is studied. In particular, it is shown that all notions coincide for non-multidimensional theories where the dimensions are associated to strongly minimal types.