Ordered trees with distinguished children

TL;DR

This paper introduces a novel ordered tree model with a distinguished child per node, deriving its enumeration via a cubic generating function and exploring various parameters, with potential for further research and extensions.

Contribution

It presents a new ordered tree model with a distinguished child, deriving its enumeration and analyzing multiple parameters, and extends the concept to marked ordered trees.

Findings

Derived a cubic generating function for the new tree model

Analyzed parameters like root degree, leaves, height, and pathlength

Extended the model to marked ordered trees with basic enumeration

Abstract

A new tree model is introduced based on ordered trees, by distinguishing exactly one child of each node that \emph{has} children. The basic enumeration leads to a cubic equation of the generating function. The extraction of its coefficients can be done using the Lagrange inversion formula. Various parameters that are commonly studied for ordered trees can also be addressed here, like degree of the root, number of leaves, number of old leaves, height, height of leftmost leaf, and pathlength. We go through these instances and leave further parameters to later research, by either the author or some readers. Dealing with cubic equations in a meaningful way requires some skills with Maple. In a last section, ordered trees are replaced by marked ordered trees; they are then combined with the concept of distinguished children. Only the basic enumeration is provided, leaving further analysis to the future.