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The Sackin and Colless indices are two widely-used metrics for measuring thebalance of trees and for testing evolutionary models in phylogenetics. Thisshort paper contributes two results about the Sackin and Colless indices oftrees. One result is the asymptotic analysis of the expected Sackin and Collessindices of a tree shape (which are full binary rooted unlabelled trees) underthe uniform model where tree shapes are sampled with equal probability. Anotheris a short elementary proof of the closed formula for the expected Sackin indexof phylogenetic trees (which are full binary rooted trees with leaves beinglabelled with taxa) under the uniform model.

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