On embedded trees and lattice paths

TL;DR

This paper applies a method for solving algebraic recurrence relations to enumerate embedded trees and lattice paths, unifying and extending existing results with new insights.

Contribution

It demonstrates how to use the method on various embedded tree families and lattice paths, providing new enumeration results and simplifying existing approaches.

Findings

Unified enumeration formulas for embedded binary trees

Extended enumeration results for embedded d-ary trees

Simplified solutions for lattice path enumeration without kernel method

Abstract

Bouttier, Di Francesco and Guitter introduced a method for solving certain classes of algebraic recurrence relations arising the context of embedded trees and map enumeration. The aim of this note is to apply this method to three problems. First, we discuss a general family of embedded binary trees, trying to unify and summarize several enumeration results for binary tree families, and also to add new results. Second, we discuss the family of embedded $d$-ary trees, embedded in the plane in a natural way. Third, we show that several enumeration problems concerning simple families of lattice paths can be solved without using the kernel method by regarding simple families of lattice paths as degenerated families of embedded trees.