Two results about the Sackin and Colless indices for phylogenetic trees and their shapes
The authors find the asymptotic behaviour of the expectation of the Sackin and Colless indices for the uniform tree model. They find that both behave as \(\pi^{1/2}\lambda^{-1}n^{3/2}\) for large \(n\), where \(n\) is the number of tips of the tree, and \(\lambda\approx 1.1300337163\), is a constant. For phylogenetic trees, under the uniform model, the expectation of the Sackin index was derived in [\textit{A. Mir} et al., Math. Biosci. 241, No. 1, 125--136 (2013; Zbl 1303.92084)] in closed form. The difference in the models is that while a \emph{phylogenetic tree} has labelled leaves, the considered in the work \textit{tree} does not. In the work here, the authors propose an alternative proof of the above closed form result for uniform phylogenetic trees. The asymptotic behaviour of the expectation of the Sackin and Colless indices for uniform phylogenetic trees is known to be also \(\pi^{1/2}n^{3/2}\).
scientific article; zbMATH DE number 7629595
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