Combinatorial differential operators in: Fa\`a di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees

We obtain a differential equation for the enumeration of the path length of general increasing trees. By using differential operators and their combinatorial interpretation we give a bijective proof of a version of Fa\`a di Bruno formula, and model the generation of ballot and Dyck paths. We get formulas for its enumeration according with the height of their lattice points. Recursive formulas for the enumeration of enriched increasing trees and forests with respect to the height of their internal and external vertices are also obtained. Finally we present a generalized form of all those results using one-parameter groups in the general context of formal power series in an arbitrary number of variables.