  Let A be a class of objects, equipped with an integer size such that for all
n the number a(n) of objects of size n is finite. We are interested in the case
where the generating fucntion sum_n a(n) t^n is rational, or more generally
algebraic. This property has a practical interest, since one can usually say a
lot on the numbers a(n), but also a combinatorial one: the rational or
algebraic nature of the generating function suggests that the objects have a
(possibly hidden) structure, similar to the linear structure of words in the
rational case, and to the branching structure of trees in the algebraic case.
We describe and illustrate this combinatorial intuition, and discuss its
validity. While it seems to be satisfactory in the rational case, it is
probably incomplete in the algebraic one. We conclude with open questions.
