Measuring tree balance using symmetry nodes - A new balance index and its extremal properties

Tree balance in phylogenetic models

Tree shape statistics, particularly measures of tree (im)balance, play an important role in the analysis of the shape of phylogenetic trees. With applications ranging from testing evolutionary models to studying the impact of fertility inheritance and selection, or tumour development and language evolution, the assessment of measures of tree balance is important. Currently, a multitude of at least 30 (im)balance indices can be found in the literature, alongside numerous other tree shape statistics. This diversity prompts essential questions: how can we assist researchers in choosing only a small number of indices to mitigate the challenges of multiple testing? Is there a preeminent balance index tailored to specific tasks? This research expands previous studies on the examination of index power, encompassing almost all established indices and a broader array of alternative models, such as a variety of trait-based models. Our investigation reveals distinct groups of balance indices better suited for different tree models, suggesting that decisions on balance index selection can be enhanced with prior knowledge. Furthermore, we present the R software package poweRbal which allows the inclusion of new indices and models, thus facilitating future research on the power of tree shape statistics.This article is part of the theme issue '"A mathematical theory of evolution": phylogenetic models dating back 100 years'.

Phylogenetic tree statistics: A systematic overview using the new R package 'treestats'

Phylogenetic trees are believed to contain a wealth of information on diversification processes. However, comparing phylogenetic trees is not straightforward due to their high dimensionality. Researchers have therefore defined a wide range of low-dimensional summary statistics. Currently, it remains unexplored to what extent these summary statistics cover the same underlying information and what summary statistics best explain observed variation across phylogenies. Furthermore, a large subset of available summary statistics focusses on measuring the topological features of a phylogenetic tree, but are often only explored at the extreme edge cases of the fully balanced or imbalanced tree and not for trees of intermediate balance. Here, we introduce a new R package called 'treestats', that provides speed optimized code to compute 70 summary statistics. We study correlations between summary statistics on empirical trees and on trees simulated using several diversification models. Furthermore, we introduce an algorithm to create intermediately balanced trees in a well-defined manner, in order to explore variation in summary statistics across a balance gradient. We find that almost all summary statistics are correlated with tree size, and find that it is difficult, if not impossible, to correct for tree size, unless the tree generating model is known. Furthermore, we find that across empirical and simulated trees, at least three large clusters of correlated summary statistics can be found, where statistics group together based on information used (topology or branching times). However, the finer grained correlation structure appears to depend strongly on either the taxonomic group studied (in empirical studies) or the tree generating model (in simulation studies). Amongst statistics describing the (im)balance of a tree, we find that almost all statistics vary non-linearly, and sometimes even non-monotonically, with our generated balance gradient. This indicates that balance is perhaps a more complex property of a tree than previously thought. Furthermore, using our new imbalancing algorithm, we devise a numerical test to identify balance statistics, and identify several statistics as balance statistics that were not previously considered as such. Lastly, our results lead to several recommendations on which statistics to select when analyzing and comparing phylogenetic trees.

The Limits of the Constant-rate Birth-Death Prior for Phylogenetic Tree Topology Inference

Birth-death models are stochastic processes describing speciation and extinction through time and across taxa and are widely used in biology for inference of evolutionary timescales. Previous research has highlighted how the expected trees under the constant-rate birth-death (crBD) model tend to differ from empirical trees, for example, with respect to the amount of phylogenetic imbalance. However, our understanding of how trees differ between the crBD model and the signal in empirical data remains incomplete. In this Point of View, we aim to expose the degree to which the crBD model differs from empirically inferred phylogenies and test the limits of the model in practice. Using a wide range of topology indices to compare crBD expectations against a comprehensive dataset of 1189 empirically estimated trees, we confirm that crBD model trees frequently differ topologically compared with empirical trees. To place this in the context of standard practice in the field, we conducted a meta-analysis for a subset of the empirical studies. When comparing studies that used Bayesian methods and crBD priors with those that used other non-crBD priors and non-Bayesian methods (i.e., maximum likelihood methods), we do not find any significant differences in tree topology inferences. To scrutinize this finding for the case of highly imbalanced trees, we selected the 100 trees with the greatest imbalance from our dataset, simulated sequence data for these tree topologies under various evolutionary rates, and re-inferred the trees under maximum likelihood and using the crBD model in a Bayesian setting. We find that when the substitution rate is low, the crBD prior results in overly balanced trees, but the tendency is negligible when substitution rates are sufficiently high. Overall, our findings demonstrate the general robustness of crBD priors across a broad range of phylogenetic inference scenarios but also highlight that empirically observed phylogenetic imbalance is highly improbable under the crBD model, leading to systematic bias in data sets with limited information content.

Phylogenetic Diversity Rankings in the Face of Extinctions: the Robustness of the Fair Proportion Index

Planning for the protection of species often involves difficult choices about which species to prioritize, given constrained resources. One way of prioritizing species is to consider their "evolutionary distinctiveness", i.e. their relative evolutionary isolation on a phylogenetic tree. Several evolutionary isolation metrics or phylogenetic diversity indices have been introduced in the literature, among them the so-called Fair Proportion index (also known as the "evolutionary distinctiveness" score). This index apportions the total diversity of a tree among all leaves, thereby providing a simple prioritization criterion for conservation. Here, we focus on the prioritization order obtained from the Fair Proportion index and analyze the effects of species extinction on this ranking. More precisely, we analyze the extent to which the ranking order may change when some species go extinct and the Fair Proportion index is re-computed for the remaining taxa. We show that for each phylogenetic tree, there are edge lengths such that the extinction of one leaf per cherry completely reverses the ranking. Moreover, we show that even if only the lowest ranked species goes extinct, the ranking order may drastically change. We end by analyzing the effects of these two extinction scenarios (extinction of the lowest ranked species and extinction of one leaf per cherry) for a collection of empirical and simulated trees. In both cases, we can observe significant changes in the prioritization orders, highlighting the empirical relevance of our theoretical findings.