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

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.