Ordered trees with distinguished children

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.

Figures (6)

• Figure 1: All 5 ordered trees with 4 nodes and the old leaves marked in red.
• Figure 2: All 5 ordered trees with nodes and each leftmost edge marked in red.
• Figure 3: All 10 ordered trees with nodes and exactly one (distinguished) child marked in red.
• Figure 4: All 10 marked ordered trees with nodes.
• Figure 5: Symbolic equation for marked ordered trees, $\mathscr{A}\cdots\mathscr{A}$ refers to $\ge0$ copies of $\mathscr{A}$.