Expected value of the Sackin index under biased speciation models.
- 1Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, 94305, USA. [email protected].
- 2Department of Biology, Stanford University, Stanford, CA, 94305, USA.
Abstract
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.
Related
- E-Backtesting Expected Shortfall without Value-at-Risk Forecasts under Time-series Models
- Calculating the Expected Value of Sample Information for Observational Studies affected by Confounding
- Nonlinear Reduced-Bias Extreme Value Index Estimators: A Comparative Study
- Prognostic value of the systemic inflammation response index (SIRI) in glioma: A meta-analysis
- Transition to an oil-based economy in Senegal: What are the expected effects on external macroeconomic dynamics?
BibTeX
@article{bauman2026expected,
title = {{Expected value of the Sackin index under biased speciation models.}},
author = {Daniel J Bauman and Noah A Rosenberg},
journal = {Journal of mathematical biology},
year = {2026},
month = oct,
doi = {10.1007/s00285-026-02465-4},
url = {https://europepmc.org/article/MED/42873318},
note = {CC BY},
}
RIS
TY - JOUR
TI - Expected value of the Sackin index under biased speciation models.
AU - Daniel J Bauman
AU - Noah A Rosenberg
PY - 2026
DA - 2026/10/14
JO - Journal of mathematical biology
DO - 10.1007/s00285-026-02465-4
UR - https://europepmc.org/article/MED/42873318
AB - 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.
SN - 0303-6812
SN - 1432-1416
ER -