Generating Functions of Embedded Trees and Lattice Paths

Bouttier, Di Francesco and Guitter introduced a method for solving certain classes of algebraic recurrence relations arising the context of maps and embedded trees. The aim of this note is to apply their method, consisting of a suitable ansatz and (computer assisted) guessing, to three problems, all related to the enumeration of lattice paths. First, we derive the generating function of a family of embedded binary trees, unifying some earlier results in the literature. Second, we show that several enumeration problems concerning so-called simple families of lattice paths can be solved without using the kernel method. Third, we use their method to (re-)derive the length generating function of three vicious walkers and osculating walkers.