Approximating algebraic functions by means of rational ones

In this paper, we use ECO method and the concept of succession rule to enumerate restricted classes of combinatorial objects. Let Ω be the succession rule describing a construction of a combinatorial objects class, then the construction of the restricted class is described by means of an approximating succession rule Ωk obtained from Ω in a natural way. We give sufficient conditions for the rule Ωk to be finite; finally we determine finite approximating rules for various classes of paths, and the approximation of the corresponding algebraic language with a regular one.