Balance indices that quantify the symmetry of branching events and the compactness of trees are widely used to compare evolutionary processes or tree-generating algorithms. Yet existing indices are not defined for all rooted trees, are unreliable for comparing trees with different numbers of leaves, and are sensitive to the presence or absence of rare types. The contributions of this article are twofold. First, we define a new class of robust, universal tree balance indices. These indices take a form similar to Colless’ index but can account for population sizes, are defined for trees with any degree distribution, and enable meaningful comparison of trees with different numbers of leaves. Second, we show that for bifurcating and all other full m-ary cladograms (in which every internal node has the same out-degree), one such Colless-like index is equivalent to the normalised reciprocal of Sackin’s index. Hence we both unify and generalise the two most popular existing tree balance indices. Our indices are intrinsically normalised and can be computed in linear time. We conclude that these more widely applicable indices have potential to supersede those in current use.

中文翻译：

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稳健、通用的树平衡指数


量化分支事件的对称性和树的紧凑性的平衡指数被广泛用于比较进化过程或树生成算法。然而，现有的指数并未针对所有有根树木进行定义，对于比较具有不同叶子数量的树木而言并不可靠，并且对稀有类型的存在或不存在敏感。本文的贡献是双重的。首先，我们定义一类新的稳健、通用的树平衡指数。这些指数采用与 Collless 指数类似的形式，但可以考虑种群规模，为具有任何程度分布的树木定义，并且能够对具有不同叶子数量的树木进行有意义的比较。其次，我们证明，对于分叉和所有其他完整的 m 分支图（其中每个内部节点具有相同的出度），一个这样的 Collless 类索引相当于 Sackin 索引的归一化倒数。因此，我们统一并概括了两个最流行的现有树木平衡指数。我们的指数本质上是标准化的，可以在线性时间内计算。我们的结论是，这些更广泛适用的指数有可能取代当前使用的指数。