Department of Biology, Stanford University, Stanford, CA 94305, United States of America.
Department of Biology, Stanford University, Stanford, CA 94305, United States of America.
Math Biosci. 2021 Dec;342:108688. doi: 10.1016/j.mbs.2021.108688. Epub 2021 Sep 16.
In mathematical phylogenetics, the Sackin index, measuring the sum of path lengths between leaves and the root, is one of the most frequently used measures of balance for phylogenetic trees. The uniform model, in which all rooted binary labeled trees for a given set of leaf labels are assumed to be equiprobable, is one of the most frequently used models for describing a probability distribution on the set of rooted binary labeled trees. This note provides a simple new derivation of the mean value of the Sackin index of tree balance under the uniform model on rooted binary labeled trees. The new derivation suggests a simple form of the mean Sackin index in terms of the Catalan numbers, quickly enabling a verification of the asymptotic value for the mean.
在数学系统发生学中，用于测量叶子和根之间路径长度总和的 Sackin 指数是衡量系统发生树平衡度的最常用指标之一。在假设所有基于给定叶子标签的有根二叉标记树的概率相等的均匀模型中，它是描述有根二叉标记树集合上概率分布的最常用模型之一。本注释提供了一种基于均匀模型下有根二叉标记树的 Sackin 指数均值的简单新推导。新的推导方法以 Catalan 数的形式给出了 Sackin 指数均值的简单形式，从而可以快速验证均值的渐近值。
