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.
