Tree balance indices: a comprehensive survey

Abstract

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.

Explicit solution of divide-and-conquer dividing by a half recurrences with polynomial independent term

“…Using this expression in Eqn. (15), and taking into account that M 1 (n) = n, we obtain the formula in the statement.…”

“…A popular set of tools used in the analysis of phylogenetic tree shapes are the shape indices [10], and among them the balance indices, which measure the propensity of the direct descendants of any given node in a tree to have the same number of descendant leaves. Many balance indices have been proposed in the literature: see, for instance, [6,12,16,17,24,25,26,27,31,33] and the recent survey [15] and the references therein. Let us recall here the four indices that appear below in the Examples section:…”

“…1). The four indices described above satisfy this condition [15]. Now, given a balance index I, it is unreasonable to expect it to allow for the comparison of the balance of trees with different numbers of leaves.…”

Two Results about the Sackin and Colless Indices for Phylogenetic Trees and Their Shapes

“…For a phylogenetic tree, its Sackin index is defined as the sum over its internal nodes of the number of leaves below that node, whereas its Colless index is defined as the sum over its internal nodes of the balance of that node, where the balance of a node is the difference in the number of leaves between the two subtrees rooted at the two children of that node. Because of their wide applications, different mathematical issues of these two tree balance metrics and other indices have been extensively studied in the past decades (see the recent comprehensive survey [10]).…”

“…Here, tree shapes (also called Otter or Polya trees) are rooted full binary trees with unlabeled leaves where each internal node has two children. To to best of our knowledge, the statistic properties of these two indices and other tree balance indices have not been formally studied [10].…”