Expected value of the Sackin index under biased speciation models

Models of random phylogenetic trees are important mathematical tools in phylogenetic data analysis; tree balance is a property of phylogenetic trees that helps describe and compare their shapes. Here, we investigate the Sackin index, one of the oldest and most widely used tree balance measures, under a variety of phylogenetic models beyond the standard Yule and Uniform models. The general biased speciation model is a biologically interpretable tree-generating model in which species branch into pairs of descendants according to an intrinsic branching rate, which is then split between the pair of descendants. The proportion of the branching rate of the parent that is assigned to a specific descendant, the split fraction, is a model parameter. Our main result is a formula for the expected value of the Sackin index under the general biased speciation model. Special cases of the general biased speciation model include a case that sets the split fraction to be a constant for each branching event, and two special cases inspired by the Aldous beta-splitting model: a random choice of split fraction is sampled independently at each branching event and drawn from a symmetric beta distribution in the Blum-François beta-splitting model or an asymmetric beta distribution in the Sainudiin-Véber beta-splitting model. As a corollary of our main result, we find formulas for the expected value of the Sackin index under these three cases of the general biased speciation model. We use the formulas to study how the balance of trees depends on the model parameters. Finally, because the Yule model is a special case of the Blum-François beta-splitting model, we find a new proof for the classical result describing the expected value of the Sackin index under the Yule model.