Measures of tree balance play an important role in the analysis of phylogenetic trees. One of the oldest and most popular indices in this regard is the Colless index for rooted bifurcating trees, introduced by Colless (Syst Zool 31:100-104, 1982). While many of its statistical properties under different probabilistic models for phylogenetic trees have already been established, little is known about its minimum value and the trees that achieve it. In this manuscript, we fill this gap in the literature. To begin with, we derive both recursive and closed expressions for the minimum Colless index of a tree with n leaves. Surprisingly, these expressions show a connection between the minimum Colless index and the so-called Blancmange curve, a fractal curve. We then fully characterize the tree shapes that achieve this minimum value and we introduce both an algorithm to generate them and a recurrence to count them. After focusing on two extremal classes of trees with minimum Colless index (the maximally balanced trees and the greedy from the bottom trees), we conclude by showing that all trees with minimum Colless index also have minimum Sackin index, another popular balance index.

中文翻译：

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关于Colless索引的最小值和达到该索引的分支树。

树木平衡的测​​量在系统发育树的分析中起着重要作用。在这方面最古老和最受欢迎的索引之一是由Colless引入的生根分叉树的Colless索引（Syst Zool 31：100-104，1982）。虽然已经建立了系统树在不同概率模型下的许多统计属性，但对其最小值和达到该最小值的树知之甚少。在这份手稿中，我们填补了文献中的空白。首先，我们导出具有n个叶子的树的最小Colless索引的递归表达式和封闭表达式。令人惊讶的是，这些表达式显示了最小的Colless指数和所谓的Blancmange曲线（分形曲线）之间的联系。然后，我们充分表征达到此最小值的树形，并介绍生成它们的算法和对它们进行计数的递归。在关注具有最低Colless指数的两类极端树（最大平衡树和底部树的贪婪度）后，我们得出结论，表明所有具有最低Colless指数的树也具有最低Sackin指数，这是另一个受欢迎的平衡指数。