A new proof is provided for the mean of the Sackin index of tree balance under the uniform model on rooted binary labeled trees.
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The new proof makes connections to results on exchangeable probability distributions and binomial identities.
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The proof simplifies the understanding of a fundamental result in the study of tree balance.
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The mean Sackin index under the uniform model can be reformulated in terms of Catalan numbers.
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This reformulation leads quickly to the asymptotic result for the mean Sackin index under the uniform model.
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.
