In the early 60s Chomsky and Schützenberger developed their mathematical hierarchy for the classification of formal languages in mathematics, computer science and linguistics. In particular, from the enumerative point of view, they proved that the first two levels of this hierarchy, rational languages and algebraic languages (aka non-ambiguous context-free languages) are closely related to rational and algebraic functions, via the notion of generating series. 

I will discuss some recent developments of these ideas relative to Schützenberger's inverse question: does a family of combinatorial structures with an $\mathbb{N}$-algebraic generating series always admit a context-free specification? While the answer is not expected to be positive in general, we have shown with Enrica Duchi that for the prominent family of combinatorial structures governed by first-order equations with discrete differential equations, a context-free specification can be derived using parking trees and a combinatorial operation of rewiring. As an illustration we will consider the enumeration of certain classes of $\lambda$-terms and of permutations.