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We obtain a differential equation for the enumeration of the path length ofgeneral increasing trees. By using differential operators and theircombinatorial interpretation we give a bijective proof of a version of Fa\`a diBruno formula, and model the generation of ballot and Dyck paths. We getformulas for its enumeration according with the height of their lattice points.Recursive formulas for the enumeration of enriched increasing trees and forestswith respect to the height of their internal and external vertices are alsoobtained. Finally we present a generalized form of all those results usingone-parameter groups in the general context of formal power series in anarbitrary number of variables.

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