Unbalanced subtrees in binary rooted ordered and un-ordered trees

TL;DR

This paper investigates the properties of unbalanced subtrees in binary rooted trees, introducing a new parameter based on Colless's index and providing enumeration results for these structures.

Contribution

It introduces the concept of the largest unbalanced subtree measured by Colless's index and offers enumeration results for these subtrees in both ordered and un-ordered binary trees.

Findings

Enumeration formulas for unbalanced subtrees

Analysis of unbalanced subtree sizes

Extension to both ordered and un-ordered trees

Abstract

Binary rooted trees, both in the ordered and in the un-ordered case, are well studied structures in the field of combinatorics. The aim of this work is to study particular patterns in these classes of trees. We consider completely unbalanced subtrees, where unbalancing is measured according to the so-called Colless's index. The size of the biggest unbalanced subtree becomes then a new parameter with respect to which we find several enumerations.