  We show that every definable subset of an uncountably categorical
pseudofinite structure has pseudofinite cardinality which is polynomial (over
the rationals) in the size of any strongly minimal subset, with the degree of
the polynomial equal to the Morley rank of the subset. From this fact, we show
that classes of finite structures whose ultraproducts all satisfy the same
uncountably categorical theory are polynomial $R$-mecs as well as
$N$-dimensional asymptotic classes, where $N$ is the Morley rank of the theory.
