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Measuring tree balance using symmetry nodes — A new balance index and its extremal properties
Mathematical Biosciences ( IF 4.3 ) Pub Date : 2021-08-22 , DOI: 10.1016/j.mbs.2021.108690
Sophie J Kersting 1 , Mareike Fischer 1
Affiliation  

Effects like selection in evolution as well as fertility inheritance in the development of populations can lead to a higher degree of asymmetry in evolutionary trees than expected under a null hypothesis. To identify and quantify such influences, various balance indices were proposed in the phylogenetic literature and have been in use for decades. However, so far no balance index was based on the number of symmetry nodes, even though symmetry nodes play an important role in other areas of mathematical phylogenetics and despite the fact that symmetry nodes are a quite natural way to measure balance or symmetry of a given tree. The aim of this manuscript is thus twofold: First, we will introduce the symmetry nodes index as an index for measuring balance of phylogenetic trees and analyze its extremal properties. We also show that this index can be calculated in linear time. This new index turns out to be a generalization of a simple and well-known balance index, namely the cherry index, as well as a specialization of another, less established, balance index, namely Rogers’ J index. Thus, it is the second objective of the present manuscript to compare the new symmetry nodes index to these two indices and to underline its advantages. In order to do so, we will derive some extremal properties of the cherry index and Rogers’ J index along the way and thus complement existing studies on these indices. Moreover, we used the programming language R to implement all three indices in the software package symmeTree, which has been made publicly available.



中文翻译:

使用对称节点测量树平衡——一种新的平衡指数及其极值特性

进化中的选择以及种群发展中的生育力遗传等效应可能导致进化树的不对称程度高于零假设下的预期。为了识别和量化这些影响,系统发育文献中提出了各种平衡指数,并且已经使用了几十年。然而,到目前为止,还没有平衡指数是基于对称节点的数量尽管对称节点在数学系统发育学的其他领域发挥着重要作用,尽管对称节点是一种非常自然的方式来衡量给定的平衡或对称性。树。因此,本手稿的目的有两个:首先,我们将介绍对称节点索引作为衡量系统发育树平衡的指标并分析其极值特性。我们还表明,该指数可以在线性时间内计算。这个新指数原来是一个简单而著名的平衡指数的概括,即樱桃指数,以及另一个不太成熟的平衡指数,即罗杰斯的 J 指数的专业化。因此,将新的对称节点索引与这两个索引进行比较并强调其优势是本手稿的第二个目标。为了做到这一点,我们将一路推导出樱桃指数和罗杰斯J指数的一些极值特性,从而补充对这些指数的现有研究。此外,我们使用了编程语言R实现软件包symmeTree中的所有三个索引,该软件包已公开。

更新日期:2021-10-15
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