Extremal Values of the Sackin Tree Balance Index

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, ones allow for efficient data structuring Therefore, methods to measure of were introduced. The Sackin index is one most frequently used measures this purpose. In many contexts, statements about minimal maximal values have discussed, but formal proofs only provided some them, context binary (search) not general rooted trees. Moreover, while number with as well when leaves power 2 relatively easy understand, all other numbers completely unknown. manuscript, we extend findings on indices from literature subsequently use our results provide formulas explicitly calculate such We also previous studies by analyzing case underlying need be binary. Finally, contribute both phylogenetic scientific using new derive non-binary settings. All implemented Mathematica package SackinMinimizer, which made publicly available.