  In this paper, we consider sequences of polynomials that satisfy
differential--difference recurrences. Our interest is motivated by the fact
that polynomials satisfying such recurrences frequently appear as generating
polynomials of integer valued random variables that are of interest in discrete
mathematics. It is, therefore, of interest to understand the properties of such
polynomials and their probabilistic consequences. As an illustration we analyze
probabilistic properties of tree--like tableaux, combinatorial objects that are
connected to asymmetric exclusion processes. In particular, we show that the
number of diagonal boxes in symmetric tree--like tableaux is asymptotically
normal and that the number of occupied corners in a random tree--like tableau
is asymptotically Poisson. This extends earlier results of Aval, Boussicault,
Nadeau, and Laborde Zubieta, respectively.
