On Sackin’s original proposal: the variance of the leaves’ depths as a phylogenetic balance index

Abstract

Background: The Sackin index S of a rooted phylogenetic tree, defined as the sum of its leaves' depths, is one of the most popular balance indices in phylogenetics, and Sackin's paper (Syst Zool 21:225-6, 1972) is usually cited as the source for this index. However, what Sackin actually proposed in his paper as a measure of the imbalance of a rooted tree was not the sum of its leaves' depths, but their "variation". This proposal was later implemented as the variance of the leaves' depths by Kirkpatrick and Slatkin in (Evolution 47:1171-81, 1993), where they also posed the problem of finding a closed formula for its expected value under the Yule model. Nowadays, Sackin's original proposal seems to have passed into oblivion in the phylogenetics literature, replaced by the index bearing his name, which, in fact, was introduced a decade later by Sokal. Results: In this paper we study the properties of the variance of the leaves' depths, V, as a balance index. Firstly, we prove that the rooted trees with n leaves and maximum V value are exactly the combs with n leaves. But although V achieves its minimum value on every space BT n of bifurcating rooted phylogenetic trees with n ≤ 183 leaves at the so-called "maximally balanced trees" with n leaves, this property fails for almost every n ≥ 184. We provide then an algorithm that finds the trees in BT n with minimum V value in time O(n log(n)). Secondly, we obtain closed formulas for the expected V value of a bifurcating rooted tree with any number n of leaves under the Yule and the uniform models and, as a by-product of the computations leading to these formulas, we also obtain closed formulas for the variance under the uniform model of the Sackin index and the total cophenetic index (Mir et al., Math Biosci 241:125-36, 2013) of a bifurcating rooted tree, as well as of their covariance, thus filling this gap in the literature. Conclusion: The phylogenetics community has been wise in preferring the sum S(T) of the leaves' depths of a phylogenetic tree T over their variance V(T) as a balance index, because the latter does not seem to capture correctly the notion of balance of large bifurcating rooted trees. But it is still a valid and useful shape index.

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“…that EPL(T) |L| 2 ) is of relevance in phylogenetic studies. In that context, the external path length of a tree is called its Sackin's index [22][23][24]. This index is used as a measure of the balance or unbalance of phylogenetic trees.…”

Exploration of the optimal modularity in assembly line design

“… Each class of ALSs with even number of input components has at least one fully symmetric graph. Fully symmetric graph is defined here as a tree in which the subtrees formed by descendants of child nodes of the same parent are identical 51 . In other words, a tree graph is fully symmetric from head to tail if its pendant subtrees are isomorphic in the sense that they have the same morphological characteristics, but it is assumed here that even though the star graph is symmetric, it is not considered as fully symmetric.…”

Revisiting Shao and Sokal's $B_2$ index of phylogenetic balance

“…In our opinion this is unfortunate, because the probabilistic interpretation of B 2 is precisely what sets it apart from other balance indices. For instance, it has already been pointed out in the literature that it is not entirely clear why the Sackin index happens to correlate with our intuition of phylogenetic (im)balance -as illustrated by the fact that Sackin's original idea had little to do with the index that came to bear his name [16].…”

“…This led us to use a different approach and consider a lesser-known -and to some extent forgottenmeasure of balance known as Shao and Sokal's B 2 index. Surprisingly given its very natural interpretation and the abundance of mathematical papers studying the properties of other balance indices [7,9,12,15,16,35,36, to name but a few], it seems that the mathematical properties of this balance index have never been studied before. Meanwhile, the current consensus in the biological literature seems to be that B 2 is not as useful as other balance indices; but on closer inspection this idea can mostly be traced to a single study [3].…”