In this paper, we illustrate a method (called the ECO method) for enumerating some classes of combinatorial objects. By means of an operator, able to satisfy two particular conditions, we give some recursive descriptions of these subclasses. We use these descriptions to deduce the functional equations satisfied by the classes' generating functions. We then illustrate some applications of the method to the whole class of plane trees, to right-leafed trees and to tip-augmented trees, and determine their generating functions according to the number of their internal nodes, the number of their leaves, their right branch length and their internal path length.
Barcucci, E., Del Lungo, A., Pergola, E., & Pinzani, R. (1998). A methodology for plane tree enumeration. Discrete Mathematics, 180(1–3), 45–64. https://doi.org/10.1016/S0012-365X(97)00122-2