Terraces in Species Tree Inference from Gene Trees

Abstract

A terrace in a phylogenetic tree space is a region where all trees contain the same set of subtrees, due to certain patterns of missing data among the taxa sampled, resulting in an identical optimality score for a given data set. This was first investigated in the context of phylogenetic tree estimation from sequence alignments using maximum likelihood (ML) and maximum parsimony (MP). The concept of terraces was later extended to the species tree inference problem from a collection of gene trees, where a set of equally optimal species trees was referred to as a "pseudo" species tree terrace. Pseudo terraces do not consider the topological proximity of the trees in terms of the induced subtrees resulting from certain patterns of missing data. In this study, we mathematically characterize species tree terraces and investigate the mathematical properties and conditions that lead multiple species trees to induce/display an identical set of locus-specific subtrees owing to missing data, therefore, as a consequence of combinatorial structure, resulting in the same optimality score. We present analytical results and combinatorial characteristics of species tree terraces. We report that species tree terraces are agnostic to gene tree topologies and the discordance therein, considering which is central to developing statistically consistent species tree estimation techniques using gene tree distributions. Therefore, we introduce and characterize a special type of gene tree topology-aware terrace which we call "peak terrace", highlight its impact and importance in species tree inference, and investigate conditions on the patterns of missing data that give rise to peak terraces.

Gentrius: identifying equally scoring trees in phylogenomics with incomplete data

“…supermatrix, supertree), Gentrius can generate its corresponding stand. Since all trees from the stand have identical score under many commonly applied objective functions ( 4 – 7 ) generating stands should be routine in phylogenetic workflow.…”

“…Thus, if the objective function to evaluate the tree quality is computed from the induced subtrees independently of the complete tree, all trees from the same stand have identical score ( 5 ). For instance, this was shown for parsimony and likelihood ( 4, 6 ) in the scope of supermatrix inference and for various criteria ( 5, 7 ) used in supertree methods. Thus, missing data can directly lead to multiple equally scoring trees ( 4 – 7 ) in widely applied phylogenomic approaches.…”

“…For instance, this was shown for parsimony and likelihood ( 4, 6 ) in the scope of supermatrix inference and for various criteria ( 5, 7 ) used in supertree methods. Thus, missing data can directly lead to multiple equally scoring trees ( 4 – 7 ) in widely applied phylogenomic approaches. However, common phylogenetic software (e.g.…”

“…One of the crucial impacts of missing data on tree inference are phylogenetic terraces – topologically distinct trees, which due to combinatorial structure and the considered scoring function, have equal scores 1 . Terraces arise for parsimony and likelihood 1,2 , as well quartet consistency 8 . Thus, they affect two popular tree inference approaches: supermatrix and supertree.…”

“…Terraces influence the efficiency of tree searches 9–11 and introduce biases to bootstrap 2 . Finally, if a best-found tree is on a large terrace 1 , ignoring this fact may lead to overconfident conclusions about evolutionary relationships 1,8,12 . Thus, the knowledge of all trees on a terrace will improve the reliability of a phylogenetic analysis.…”