Extremal values of the Sackin tree balance index

TL;DR

This paper investigates the extremal values of the Sackin tree balance index for both binary and non-binary rooted trees, providing formulas for counting trees with minimal and maximal index values, and implementing these in a Mathematica package.

Contribution

It extends previous results on Sackin index extremal trees to general rooted trees and derives explicit formulas for their counts, enhancing understanding in phylogenetics and computer science.

Findings

Formulas for counting trees with minimal and maximal Sackin index.

Extension of results to non-binary trees.

Implementation of findings in the SackinMinimizer Mathematica package.

Abstract

Tree balance plays an important role in different research areas like theoretical computer science and mathematical phylogenetics. For example, it has long been known that under the Yule model, a pure birth process, imbalanced trees are more likely than balanced ones. Also, concerning ordered search trees, more balanced ones allow for more efficient data structuring than imbalanced ones. Therefore, different methods to measure the balance of trees were introduced. The Sackin index is one of the most frequently used measures for this purpose. In many contexts, statements about the minimal and maximal values of this index have been discussed, but formal proofs have only been provided for some of them, and only in the context of ordered binary (search) trees, not for general rooted trees. Moreover, while the number of trees with maximal Sackin index as well as the number of trees with minimal Sackin index when the number of leaves is a power of 2 are relatively easy to understand, the number of trees with minimal Sackin index for all other numbers of leaves has been completely unknown. In this manuscript, we extend the findings on trees with minimal and maximal Sackin indices from the literature on ordered trees and subsequently use our results to provide formulas to explicitly calculate the numbers of such trees. We also extend previous studies by analyzing the case when the underlying trees need not be binary. Finally, we use our results to contribute both to the phylogenetic as well as the computer scientific literature by using the new findings on Sackin minimal and maximal trees in order to derive formulas to calculate the number of both minimal and maximal phylogenetic trees as well as minimal and maximal ordered trees both in the binary and non-binary settings. All our results have been implemented in the Mathematica package SackinMinimizer, which has been made publicly available.