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. 2021 Dec:342:108688.
doi: 10.1016/j.mbs.2021.108688. Epub 2021 Sep 16.

A simple derivation of the mean of the Sackin index of tree balance under the uniform model on rooted binary labeled trees

Matthew C King  1 , Noah A Rosenberg  2
Affiliations

A simple derivation of the mean of the Sackin index of tree balance under the uniform model on rooted binary labeled trees

Matthew C King et al. Math Biosci. 2021 Dec.

Abstract

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.

Keywords: Catalan numbers; Phylogenetics; Tree balance.

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Conflict of interest statement

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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