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 (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.

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关于 Colless 指数的最小值和实现它的分叉树

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指数之一是 Colless (1982) 引入的用于有根分叉树的 Colless 指数。虽然它在不同的系统发育树概率模型下的许多统计特性已经建立，但对其最小值和实现它的树知之甚少。在这份手稿中，我们填补了文献中的这一空白。首先，我们推导出具有 $n$ 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 Colless 指数与所谓的 Blancmange 曲线（一种分形曲线）之间的联系。然后，我们完全表征了达到此最小值的树形状，并引入了生成它们的算法和计算它们的循环。在关注具有最小 Colless 指数的两个极值类树（最大平衡树和来自底部树的贪婪）之后，我们得出结论，所有具有最小 Colless 指数的树也具有最小 Sackin 指数，这是另一种流行的平衡指数。