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.