Darío GarcíaMelissa Robles2026-03-222026-03-22202510.1017/jsl.2025.10172https://doi.org/10.1017/jsl.2025.10172https://andeanlibrary.org/handle/123456789/78611Abstract 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.enUltraproductMathematicsRank (graph theory)Characterization (materials science)Tree (set theory)Discrete mathematicsPolynomialPure mathematicsAlgebra over a fieldAlmost everywherearticle