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 using the new findings on Sackin minimal and maximal trees 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.

树平衡在理论计算机科学和数学系统发育学等不同研究领域发挥着重要作用。例如，人们早就知道，在纯出生过程的 Yule 模型下，不平衡的树木比平衡的树木更有可能。此外，关于有序搜索树，更平衡的搜索树比不平衡的搜索树允许更有效的数据结构。因此，引入了测量树木平衡的不同方法。Sackin 指数是为此目的最常用的衡量标准之一。在许多上下文中，已经讨论了有关该索引的最小值和最大值的陈述，但仅针对其中一些提供了正式证明，并且仅在有序二叉（搜索）树的上下文中，而不是针对一般有根树。而且，而当叶子数为 2 的幂时，Sackin 指数最大的树的数量以及 Sackin 指数最小的树的数量相对容易理解，而所有其他叶子数的 Sackin 指数最小的树的数量已经完全未知。在这份手稿中，我们扩展了有关有序树文献中具有最小和最大 Sackin 指数的树的发现，然后使用我们的结果提供公式来明确计算此类树的数量。我们还通过分析底层树不需要是二元树的情况来扩展以前的研究。最后，我们使用我们的结果为系统发育和计算机科学文献做出贡献，使用 Sackin 最小和最大树的新发现来推导公式来计算最小和最大系统发育树以及最小和最大有序树的数量在二进制和非二进制设置中。我们所有的结果都在 Mathematica 包 SackinMinimizer 中实现，该包已公开可用。