# On two subclasses of Motzkin paths and their relation to ternary trees

## TL;DR

This paper introduces two subclasses of Motzkin paths, establishes bijections with ternary and non-crossing trees, and derives generating functions to analyze their combinatorial properties.

## Contribution

It presents new subclasses of Motzkin paths and their explicit bijections with ternary and non-crossing trees, along with symbolic equations and generating functions.

## Key findings

- Bijections between S-Motzkin paths and ternary trees
- Bijections between S-Motzkin paths and non-crossing trees
- Bijections between T-Motzkin paths and pairs of ternary trees

## Abstract

Two subclasses of Motzkin paths, S-Motzkin and T-Motzkin paths, are introduced. We provide bijections between S-Motzkin paths and ternary trees, S-Motzkin paths and non-crossing trees, and T-Motzkin paths and ordered pairs of ternary trees. Symbolic equations for both paths, and thus generating functions for the paths, are provided. Using these, various parameters involving the two paths are analyzed.