Extremal Values of the Sackin Tree Balance Index

Fischer, Mareike

doi:10.1007/s00026-021-00539-2

Cited by 12 publications

(17 citation statements)

References 21 publications

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“…The analyses concerning extremal values and trees for mCI (and thus also for the classic cherry index) and J nicely complement recent research on extremal properties of other classic balance indices [4,23]. The results on the J index can be directly derived from insights into the new SN I index, as we can surprisingly show that, while these indices are substantially very different, their extremal behavior completely coincides.…”

Section: Introductionsupporting

confidence: 76%

“…For decades, various balance indices have been used and discussed as a tool to measure the degree of asymmetry in trees like genealogies or phylogenies that are, for instance, reconstructed based on genetical data (see, for example, [1,2,3,4,5,6]). Their application can give insights into diversification rate variation [7], influences like fertility inheritance and selection [8,9] or effects of different tree reconstruction methods [10,11,12].…”

Section: Introductionmentioning

confidence: 99%

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Measuring tree balance using symmetry nodes -- a new balance index and its extremal properties

Kersting¹,

Fischer²

2021

Preprint

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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.

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Section: Introductionsupporting

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Measuring tree balance using symmetry nodes -- a new balance index and its extremal properties

Kersting¹,

Fischer²

2021

Preprint

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“…In this subsection, we will provide some additional results on the Sackin index and the average leaf depth that -to our knowledge -have not yet been known or not yet been proven. For that, recall that the Sackin index [27,73] of a tree T ∈ T * n is defined as 19…”

Section: A11 Average Leaf Depth and Sackin Indexmentioning

confidence: 99%

Tree balance indices: a comprehensive survey

Fischer¹,

Herbst²,

Kersting³

et al. 2021

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Tree balance plays an important role in phylogenetics and other research areas, which is why several indices to measure tree balance have been introduced over the years. Nevertheless, a formal definition of what a balance index actually is and what makes it a useful measure of balance (or, in other cases, imbalance), has so far not been introduced in the literature. While the established indices all summarize the (im)balance of a tree in a single number, they vary in their definitions and underlying principles. It is the aim of the present manuscript to introduce formal definitions of balance and imbalance indices that classify desirable properties of such indices and to analyze and categorize established indices accordingly.In this regard, we review 19 established (im)balance indices from the literature, summarize their general, statistical and combinatorial properties (where known), prove numerous additional results and indicate directions for future research by making explicit open questions and gaps in the literature. We also prove that a few tree shape statistics that have been used to measure tree balance in the literature do not fulfill our definition of an (im)balance index, which might indicate that their properties are not as useful for practical purposes. Moreover, we show that five additional tree shape statistics from other contexts actually are tree (im)balance indices according to our definition. The manuscript is accompanied by the website treebalance.wordpress.com containing fact sheets of the discussed indices. Moreover, we introduce the software package treebalance implemented in R that can be used to calculate all indices discussed.

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“…By showing that all trees with minimal Colless index also have minimal Sackin index, we derive an upper bound for this number, since the number of trees with minimal Sackin index is known in the literature (cf. Theorem 8 in Fischer (2018)). Using some knowledge on the leaf partitioning induced by the root of trees with minimum Colless index, we are able to improve this bound even further.…”

Section: Introductionmentioning

confidence: 99%

“…Definition 3 on page 31) we have S(T 1 ) = S(T 2 ) = 44, which can be shown to be minimal (cf. Theorem 3 inFischer (2018)).…”

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confidence: 99%

Extremal properties of the Colless balance index for rooted binary trees

Fischer,

Herbst,

Wicke

2019

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Measures of tree balance play an important role in various research areas, for example in phylogenetics. There they are for instance used to test whether an observed phylogenetic tree differs significantly from a tree generated by the Yule model of speciation. One of the most popular indices in this regard is the Colless index, which measures the degree of balance for rooted binary trees. While many statistical properties of the Colless index (e.g. asymptotic results for its mean and variance under different models of speciation) have already been discussed in different contexts, we focus on its extremal properties. While it is relatively straightforward to characterize trees with maximal Colless index, the analysis of the minimal value of the Colless index and the characterization of trees that achieve it, are much more involved. In this note, we therefore focus on the minimal value of the Colless index for any given number of leaves. We derive both a recursive formula for this minimal value, as well as an explicit expression, which shows a surprising connection between the Colless index and the so-called Blancmange curve, a fractal curve that is also known as the Takagi curve. Moreover, we characterize two classes of trees that have minimal Colless index, consisting of the set of so-called maximally balanced trees and a class of trees that we call greedy from the bottom trees. Furthermore, we derive an upper bound for the number of trees with minimal Colless index by relating these trees with trees with minimal Sackin index (another well-studied index of tree balance).

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Extremal Values of the Sackin Tree Balance Index

Cited by 12 publications

References 21 publications

Measuring tree balance using symmetry nodes -- a new balance index and its extremal properties

Measuring tree balance using symmetry nodes -- a new balance index and its extremal properties

Tree balance indices: a comprehensive survey

Extremal properties of the Colless balance index for rooted binary trees

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