Set Of Gene Trees Research Articles

MUL-tree pruning for consistency and optimal reconciliation - complexity and algorithms

A multi-labeled tree (or MUL-tree) is a rooted tree in which every leaf is labeled by an element from some set X, but in which more than one leaf may be labeled by the same element of X. MUL-trees have applications in many fields. In phylogenetics, they can represent the evolution of gene families, where genes are represented by the species they belong to, the non-uniqueness of leaf-labels coming from the fact that a given genome may contain many paralogous genes (genes related through duplication). In this paper, we consider two problems related to the leaf-pruning (leaf removal) of MUL-trees leading to single-labeled trees. First, given a set of MUL-trees, the MUL-tree Set Pruning for Consistency (MULSETPC) Problem asks for a pruning of each tree leading to a set of consistent trees, i.e. a collection of label-isomorphic single-labeled trees. Second, processing one tree at a time, the MUL-tree Pruning for Reconciliation (MULPR) Problem asks for a pruning that minimizes a “reconciliation” cost, i.e. an evolutionary cost of the gene family through duplication, given a known species tree. We show that MULSETPC is NP-hard and that MULPR is W[2]-hard when parameterized by the duplication cost, NP-hard when the number of occurrences of each label in a MUL-tree is bounded by two and that the optimization version of MULPR is hard to approximate within factor (1−o(1))ln⁡|X|. We then develop a polynomial-time heuristic for MULPR and show its accuracy by comparing it to a brute-force exact method on a set of gene trees from the Ensembl Genome Browser.

Completing gene trees without species trees in sub-quadratic time.

As genome-wide reconstruction of phylogenetic trees becomes more widespread, limitations of available data are being appreciated more than ever before. One issue is that phylogenomic datasets are riddled with missing data, and gene trees, in particular, almost always lack representatives from some species otherwise available in the dataset. Since many downstream applications of gene trees require or can benefit from access to complete gene trees, it will be beneficial to algorithmically complete gene trees. Also, gene trees are often unrooted, and rooting them is useful for downstream applications. While completing and rooting a gene tree with respect to a given species tree has been studied, those problems are not studied in depth when we lack such a reference species tree. We study completion of gene trees without a need for a reference species tree. We formulate an optimization problem to complete the gene trees while minimizing their quartet distance to the given set of gene trees. We extend a seminal algorithm by Brodal et al. to solve this problem in quasi-linear time. In simulated studies and on a large empirical data, we show that completion of gene trees using other gene trees is relatively accurate and, unlike the case where a species tree is available, is unbiased. Our method, tripVote, is available at https://github.com/uym2/tripVote. Supplementary data are available at Bioinformatics online.

Species Tree Estimation from Gene Trees by Minimizing Deep Coalescence and Maximizing Quartet Consistency: A Comparative Study and the Presence of Pseudo Species Tree Terraces.

Species tree estimation from multilocus data sets is extremely challenging, especially in the presence of gene tree heterogeneity across the genome due to incomplete lineage sorting (ILS). Summary methods have been developed which estimate gene trees and then combine the gene trees to estimate a species tree by optimizing various optimization scores. In this study, we have extended and adapted the concept of phylogenetic terraces to species tree estimation by "summarizing" a set of gene trees, where multiple species trees with distinct topologies may have exactly the same optimality score (i.e., quartet score, extra lineage score, etc.). We particularly investigated the presence and impacts of equally optimal trees in species tree estimation from multilocus data using summary methods by taking ILS into account. We analyzed two of the most popular ILS-aware optimization criteria: maximize quartet consistency (MQC) and minimize deep coalescence (MDC). Methods based on MQC are provably statistically consistent, whereas MDC is not a consistent criterion for species tree estimation. We present a comprehensive comparative study of these two optimality criteria. Our experiments, on a collection of data sets simulated under ILS, indicate that MDC may result in competitive or identical quartet consistency score as MQC, but could be significantly worse than MQC in terms of tree accuracy-demonstrating the presence and impacts of equally optimal species trees. This is the first known study that provides the conditions for the data sets to have equally optimal trees in the context of phylogenomic inference using summary methods. [Gene tree; incomplete lineage sorting; phylogenomic analysis, species tree; summary method.].

A new fast method for inferring multiple consensus trees using k-medoids

BackgroundGene trees carry important information about specific evolutionary patterns which characterize the evolution of the corresponding gene families. However, a reliable species consensus tree cannot be inferred from a multiple sequence alignment of a single gene family or from the concatenation of alignments corresponding to gene families having different evolutionary histories. These evolutionary histories can be quite different due to horizontal transfer events or to ancient gene duplications which cause the emergence of paralogs within a genome. Many methods have been proposed to infer a single consensus tree from a collection of gene trees. Still, the application of these tree merging methods can lead to the loss of specific evolutionary patterns which characterize some gene families or some groups of gene families. Thus, the problem of inferring multiple consensus trees from a given set of gene trees becomes relevant.ResultsWe describe a new fast method for inferring multiple consensus trees from a given set of phylogenetic trees (i.e. additive trees or X-trees) defined on the same set of species (i.e. objects or taxa). The traditional consensus approach yields a single consensus tree. We use the popular k-medoids partitioning algorithm to divide a given set of trees into several clusters of trees. We propose novel versions of the well-known Silhouette and Caliński-Harabasz cluster validity indices that are adapted for tree clustering with k-medoids. The efficiency of the new method was assessed using both synthetic and real data, such as a well-known phylogenetic dataset consisting of 47 gene trees inferred for 14 archaeal organisms.ConclusionsThe method described here allows inference of multiple consensus trees from a given set of gene trees. It can be used to identify groups of gene trees having similar intragroup and different intergroup evolutionary histories. The main advantage of our method is that it is much faster than the existing tree clustering approaches, while providing similar or better clustering results in most cases. This makes it particularly well suited for the analysis of large genomic and phylogenetic datasets.

Gene-Tree Reconciliation with MUL-Trees to Resolve Polyploidy Events.

Polyploidy can have a huge impact on the evolution of species, and it is a common occurrence, especially in plants. The two types of polyploids-autopolyploids and allopolyploids-differ in the level of divergence between the genes that are brought together in the new polyploid lineage. Because allopolyploids are formed via hybridization, the homoeologous copies of genes within them are at least as divergent as orthologs in the parental species that came together to form them. This means that common methods for estimating the parental lineages of allopolyploidy events are not accurate, and can lead to incorrect inferences about the number of gene duplications and losses. Here, we have adapted an algorithm for topology-based gene-tree reconciliation to work with multi-labeled trees (MUL-trees). By definition, MUL-trees have some tips with identical labels, which makes them a natural representation of the genomes of polyploids. Using this new reconciliation algorithm we can: accurately place allopolyploidy events on a phylogeny, identify the parental lineages that hybridized to form allopolyploids, distinguish between allo-, auto-, and (in most cases) no polyploidy, and correctly count the number of duplications and losses in a set of gene trees. We validate our method using gene trees simulated with and without polyploidy, and revisit the history of polyploidy in data from the clades including both baker's yeast and bread wheat. Our re-analysis of the yeast data confirms the allopolyploid origin and parental lineages previously identified for this group. The method presented here should find wide use in the growing number of genomes from species with a history of polyploidy. [Polyploidy; reconciliation; whole-genome duplication.].

PHRAPL: Phylogeographic Inference Using Approximate Likelihoods

The demographic history of most species is complex, with multiple evolutionary processes combining to shape the observed patterns of genetic diversity. To infer this history, the discipline of phylogeography has (to date) used models that simplify the historical demography of the focal organism, for example by assuming or ignoring ongoing gene flow between populations or by requiring a priori specification of divergence history. Since no single model incorporates every possible evolutionary process, researchers rely on intuition to choose the models that they use to analyze their data. Here, we describe an approximate likelihood approach that reduces this reliance on intuition. PHRAPL allows users to calculate the probability of a large number of complex demographic histories given a set of gene trees, enabling them to identify the most likely underlying model and estimate parameters for a given system. Available model parameters include coalescence time among populations or species, gene flow, and population size. We describe the method and test its performance in model selection and parameter estimation using simulated data. We also compare model probabilities estimated using our approximate likelihood method to those obtained using standard analytical likelihood. The method performs well under a wide range of scenarios, although this is sometimes contingent on sampling many loci. In most scenarios, as long as there are enough loci and if divergence among populations is sufficiently deep, PHRAPL can return the true model in nearly all simulated replicates. Parameter estimates from the method are also generally accurate in most cases. PHRAPL is a valuable new method for phylogeographic model selection and will be particularly useful as a tool to more extensively explore demographic model space than is typically done or to estimate parameters for complex models that are not readily implemented using current methods. Estimating relevant parameters using the most appropriate demographic model can help to sharpen our understanding of the evolutionary processes giving rise to phylogeographic patterns. [AIC; grid search; isolation-with-migration; migration rate; multispecies coalescent; parameter optimization; population genetics; tree topologies.].

An analytical upper bound on the number of loci required for all splits of a species tree to appear in a set of gene trees.

BackgroundMany methods for species tree inference require data from a sufficiently large sample of genomic loci in order to produce accurate estimates. However, few studies have attempted to use analytical theory to quantify “sufficiently large”.ResultsUsing the multispecies coalescent model, we report a general analytical upper bound on the number of gene trees n required such that with probability q, each bipartition of a species tree is represented at least once in a set of n random gene trees. This bound employs a formula that is straightforward to compute, depends only on the minimum internal branch length of the species tree and the number of taxa, and applies irrespective of the species tree topology. Using simulations, we investigate numerical properties of the bound as well as its accuracy under the multispecies coalescent.ConclusionsOur results are helpful for conservatively bounding the number of gene trees required by the ASTRAL inference method, and the approach has potential to be extended to bound other properties of gene tree sets under the model.

Normalizing Kernels in the Billera-Holmes-Vogtmann Treespace.

As costs of genome sequencing have dropped precipitously, development of efficient bioinformatic methods to analyze genome structure and evolution have become ever more urgent. For example, most published phylogenomic studies involve either massive concatenation of sequences, or informal comparisons of phylogenies inferred on a small subset of orthologous genes, neither of which provides a comprehensive overview of evolution or systematic identification of genes with unusual and interesting evolution (e.g., horizontal gene transfers, gene duplication, and subsequent neofunctionalization). We are interested in identifying such "outlying" gene trees from the set of gene trees and estimating the distribution of trees over the "tree space". This paper describes an improvement to the kdetrees algorithm, an adaptation of classical kernel density estimation to the metric space of phylogenetic trees (Billera-Holmes-Vogtman treespace), whereby the kernel normalizing constants, are estimated through the use of the novel holonomic gradient methods. As in the original kdetrees paper, we have applied kdetrees to a set of Apicomplexa genes. The analysis identified several unreliable sequence alignments that had escaped previous detection, as well as a gene independently reported as a possible case of horizontal gene transfer. The updated version of the kdetrees software package is available both from CRAN (the official R package system), as well as from the official development repository on Github. ( github.com/grady/kdetrees).

Fast Coalescent-Based Computation of Local Branch Support from Quartet Frequencies.

Species tree reconstruction is complicated by effects of incomplete lineage sorting, commonly modeled by the multi-species coalescent model (MSC). While there has been substantial progress in developing methods that estimate a species tree given a collection of gene trees, less attention has been paid to fast and accurate methods of quantifying support. In this article, we propose a fast algorithm to compute quartet-based support for each branch of a given species tree with regard to a given set of gene trees. We then show how the quartet support can be used in the context of the MSC to compute (1) the local posterior probability (PP) that the branch is in the species tree and (2) the length of the branch in coalescent units. We evaluate the precision and recall of the local PP on a wide set of simulated and biological datasets, and show that it has very high precision and improved recall compared with multi-locus bootstrapping. The estimated branch lengths are highly accurate when gene tree estimation error is low, but are underestimated when gene tree estimation error increases. Computation of both the branch length and local PP is implemented as new features in ASTRAL.

Unconstrained Diameters for Deep Coalescence.

The minimizing-deep-coalescence (MDC) approach infers a median (species) tree for a given set of gene trees under the deep coalescence cost. This cost accounts for the minimum number of deep coalescences needed to reconcile a gene tree with a species tree where the leaf-genes are mapped to the leaf-species through a function called leaf labeling. In order to better understand the MDC approach we investigate here the diameter of a gene tree, which is an important property of the deep coalescence cost. This diameter is the maximal deep coalescence costs for a given gene tree under all leaf labelings for each possible species tree topology. While we prove that this diameter is generally infinite, this result relies on the diameter's unrealistic assumption that species trees can be of infinite size. Providing a more practical definition, we introduce a natural extension of the gene tree diameter that constrains the species tree size by a given constant. For this new diameter, we describe an exact formula, present a complete classification of the trees yielding this diameter, derive formulas for its mean and variance, and demonstrate its ability using comparative studies.

Species Tree Inference Using a Mixture Model.

Species tree reconstruction has been a subject of substantial research due to its central role across biology and medicine. A species tree is often reconstructed using a set of gene trees or by directly using sequence data. In either of these cases, one of the main confounding phenomena is the discordance between a species tree and a gene tree due to evolutionary events such as duplications and losses. Probabilistic methods can resolve the discordance by coestimating gene trees and the species tree but this approach poses a scalability problem for larger data sets. We present MixTreEM-DLRS: A two-phase approach for reconstructing a species tree in the presence of gene duplications and losses. In the first phase, MixTreEM, a novel structural expectation maximization algorithm based on a mixture model is used to reconstruct a set of candidate species trees, given sequence data for monocopy gene families from the genomes under study. In the second phase, PrIME-DLRS, a method based on the DLRS model (Åkerborg O, Sennblad B, Arvestad L, Lagergren J. 2009. Simultaneous Bayesian gene tree reconstruction and reconciliation analysis. Proc Natl Acad Sci U S A. 106(14):5714-5719), is used for selecting the best species tree. PrIME-DLRS can handle multicopy gene families since DLRS, apart from modeling sequence evolution, models gene duplication and loss using a gene evolution model (Arvestad L, Lagergren J, Sennblad B. 2009. The gene evolution model and computing its associated probabilities. J ACM. 56(2):1-44). We evaluate MixTreEM-DLRS using synthetic and biological data, and compare its performance with a recent genome-scale species tree reconstruction method PHYLDOG (Boussau B, Szöllősi GJ, Duret L, Gouy M, Tannier E, Daubin V. 2013. Genome-scale coestimation of species and gene trees. Genome Res. 23(2):323-330) as well as with a fast parsimony-based algorithm Duptree (Wehe A, Bansal MS, Burleigh JG, Eulenstein O. 2008. Duptree: a program for large-scale phylogenetic analyses using gene tree parsimony. Bioinformatics 24(13):1540-1541). Our method is competitive with PHYLDOG in terms of accuracy and runs significantly faster and our method outperforms Duptree in accuracy. The analysis constituted by MixTreEM without DLRS may also be used for selecting the target species tree, yielding a fast and yet accurate algorithm for larger data sets. MixTreEM is freely available at http://prime.scilifelab.se/mixtreem/.

Complexity insights of the Minimum Duplication problem

The Minimum Duplication problem is a well-known problem in phylogenetics and comparative genomics. Given a set of gene trees, the Minimum Duplication problem asks for a species tree that induces the minimum number of gene duplications in the input gene trees. Recently, a variant of the Minimum Duplication problem, called Minimum Duplication Bipartite, has been introduced, where the goal is to find all pre-duplications, that is duplications that in the evolution precede the first speciation with respect to a species tree. In this paper, we investigate the complexity of both Minimum Duplication and Minimum Duplication Bipartite. First of all, we prove that the Minimum Duplication problem is APX-hard, even when the input consists of five uniquely leaf-labeled gene trees (improving upon known results on the complexity of the problem). Then, we show that the Minimum Duplication Bipartite problem can be solved efficiently with a randomized algorithm when the input gene trees have bounded depth. An extended abstract of this paper appeared in SOFSEM 2012 [1].

An algorithm for constructing parsimonious hybridization networks with multiple phylogenetic trees.

A phylogenetic network is a model for reticulate evolution. A hybridization network is one type of phylogenetic network for a set of discordant gene trees and "displays" each gene tree. A central computational problem on hybridization networks is: given a set of gene trees, reconstruct the minimum (i.e., most parsimonious) hybridization network that displays each given gene tree. This problem is known to be NP-hard, and existing approaches for this problem are either heuristics or making simplifying assumptions (e.g., work with only two input trees or assume some topological properties). In this article, we develop an exact algorithm (called PIRNC) for inferring the minimum hybridization networks from multiple gene trees. The PIRNC algorithm does not rely on structural assumptions (e.g., the so-called galled networks). To the best of our knowledge, PIRNC is the first exact algorithm implemented for this formulation. When the number of reticulation events is relatively small (say, four or fewer), PIRNC runs reasonably efficient even for moderately large datasets. For building more complex networks, we also develop a heuristic version of PIRNC called PIRNCH. Simulation shows that PIRNCH usually produces networks with fewer reticulation events than those by an existing method. PIRNC and PIRNCH have been implemented as part of the software package called PIRN and is available online.

Gene tree correction for reconciliation and species tree inference: Complexity and algorithms

Reconciliation consists in mapping a gene tree T into a species tree S, and explaining the incongruence between the two as evidence for duplication, loss and other events shaping the gene family represented by the leaves of T. When S is unknown, the Species Tree Inference Problem is to infer, from a set of gene trees, a species tree leading to a minimum reconciliation cost. As reconciliation is very sensitive to errors in T, gene tree correction prior to reconciliation is a fundamental task. In this paper, we investigate the complexity of four different combinatorial approaches for deleting misplaced leaves from T. First, we consider two problems (Minimum Leaf Removal and Minimum Species Removal) related to the reconciliation of T with a known species tree S. In the former (latter respectively) we want to remove the minimum number of leaves (species respectively) so that T is “MD-consistent” with S. Second, we consider two problems (Minimum Leaf Removal Inference and Minimum Species Removal Inference) related to species tree inference. In the former (latter respectively) we want to remove the minimum number of leaves (species respectively) from T so that there exists a species tree S such that T is MD-consistent with S. We prove that Minimum Leaf Removal and Minimum Species Removal are APX-hard, even when each label has at most two occurrences in the input gene tree, and we present fixed-parameter algorithms for the two problems. We prove that Minimum Leaf Removal Inference is not only NP-hard, but also W[2]-hard and inapproximable within factor clnn, where n is the number of leaves in the gene tree. Finally, we show that Minimum Species Removal Inference is NP-hard and W[2]-hard, when parameterized by the size of the solution, that is the minimum number of species removals.

Multiple consensus trees: a method to separate divergent genes

BackgroundIt is generally admitted that the species tree cannot be inferred from the genetic sequences of a single gene because the evolution of different genes, and thus the gene tree topologies, may vary substantially. Gene trees can differ, for example, because of horizontal transfer events or because some of them correspond to paralogous instead of orthologous sequences. A variety of methods has been proposed to tackle the problem of the reconciliation of gene trees in order to reconstruct a species tree. When the taxa in all the trees are identical, the problem can be stated as a consensus tree problem.ResultsIn this paper we define a new method for deciding whether a unique consensus tree or multiple consensus trees can best represent a set of given phylogenetic trees. If the given trees are all congruent, they should be compatible into a single consensus tree. Otherwise, several consensus trees corresponding to divergent genetic patterns can be identified. We introduce a method optimizing the generalized score, over a set of tree partitions in order to decide whether the given set of gene trees is homogeneous or not.ConclusionsThe proposed method has been validated with simulated data (random trees organized in three topological groups) as well as with real data (bootstrap trees, homogeneous set of trees, and a set of non homogeneous gene trees of 30 E. Coli strains; it is worth noting that some of the latter genes underwent horizontal gene transfers). A computer program, MCT - Multiple Consensus Trees, written in C was made freely available for the research community (it can be downloaded from http://bioinformatics.lif.univ-mrs.fr/consensus/index.html). It handles trees in a standard Newick format, builds three hierarchies corresponding to RF and QS similarities between trees and the greedy ascending algorithm. The generalized score values of all tree partitions are computed.

Evolution of gene neighborhoods within reconciled phylogenies

Motivation: Most models of genome evolution integrating gene duplications, losses and chromosomal rearrangements are computationally intract able, even when comparing only two genomes. This prevents large-scale studies that consider different types of genome structural variations.Results: We define an ‘adjacency phylogenetic tree’ that describes the evolution of an adjacency, a neighborhood relation between two genes, by speciation, duplication or loss of one or both genes, and rearrangement. We describe an algorithm that, given a species tree and a set of gene trees where the leaves are connected by adjacencies, computes an adjacency forest that minimizes the number of gains and breakages of adjacencies (caused by rearrangements) and runs in polynomial time. We use this algorithm to reconstruct contiguous regions of mammalian and plant ancestral genomes in a few minutes for a dozen species and several thousand genes. We show that this method yields reduced conflict between ancestral adjacencies. We detect duplications involving several genes and compare the different modes of evolution between phyla and among lineages.Availability: C++ implementation using BIO++ package, available upon request to Sèverine Bérard.Contact: Severine.Berard@cirad.fr or Eric.Tannier@inria.frSupplementary information: Supplementary material is available at Bioinformatics online.

A multilocus perspective on the speciation history of a North American aridland toad (Anaxyrus punctatus)

Interpretations of phylogeographic patterns can change when analyses shift from single gene-tree to multilocus coalescent analyses. Using multilocus coalescent approaches, a species tree and divergence times can be estimated from a set of gene trees while accounting for gene-tree stochasticity. We utilized the conceptual strengths of a multilocus coalescent approach coupled with complete range-wide sampling to examine the speciation history of a broadly distributed, North American warm-desert toad, Anaxyrus punctatus. Phylogenetic analyses provided strong support for three major lineages within A. punctatus. Each lineage broadly corresponded to one of three desert regions. Early speciation in A. punctatus appeared linked to late Miocene–Pliocene development of the Baja California peninsula. This event was likely followed by a Pleistocene divergence associated with the separation of the Chihuahuan and Sonoran Deserts. Our multilocus coalescent-based reconstruction provides an informative contrast to previous single gene-tree estimates of the evolutionary history of A. punctatus.

Structural properties of the reconciliation space and their applications in enumerating nearly-optimal reconciliations between a gene tree and a species tree

IntroductionA gene tree for a gene family is often discordant with the containing species tree because of its complex evolutionary course during which gene duplication, gene loss and incomplete lineage sorting events might occur. Hence, it is of great challenge to infer the containing species tree from a set of gene trees. One common approach to this inference problem is through gene tree and species tree reconciliation.ResultsIn this paper, we generalize the traditional least common ancestor (LCA) reconciliation to define a reconciliation between a gene tree and species tree under the tree homomorphism framework. We then study the structural properties of the space of all reconciliations between a gene tree and a species tree in terms of the gene duplication, gene loss or deep coalescence costs. As application, we show that the LCA reconciliation is the unique one that has the minimum deep coalescence cost, provide a novel characterization of the reconciliations with the optimal duplication cost, and present efficient algorithms for enumerating (nearly-)optimal reconciliations with respect to each cost.ConclusionsThis work provides a new graph-theoretic framework for studying gene tree and species tree reconciliations.

Detecting Highways of Horizontal Gene Transfer

In a horizontal gene transfer (HGT) event, a gene is transferred between two species that do not have an ancestor-descendant relationship. Typically, no more than a few genes are horizontally transferred between any two species. However, several studies identified pairs of species between which many different genes were horizontally transferred. Such a pair is said to be linked by a highway of gene sharing. We present a method for inferring such highways. Our method is based on the fact that the evolutionary histories of horizontally transferred genes disagree with the corresponding species phylogeny. Specifically, given a set of gene trees and a trusted rooted species tree, each gene tree is first decomposed into its constituent quartet trees and the quartets that are inconsistent with the species tree are identified. Our method finds a pair of species such that a highway between them explains the largest (normalized) fraction of inconsistent quartets. For a problem on n species and m input quartet trees, we give an efficient O(m + n(2))-time algorithm for detecting highways, which is optimal with respect to the quartets input size. An application of our method to a dataset of 1128 genes from 11 cyanobacterial species, as well as to simulated datasets, illustrates the efficacy of our method.

Consistency Properties of Species Tree Inference by Minimizing Deep Coalescences

Methods for inferring species trees from sets of gene trees need to account for the possibility of discordance among the gene trees. Assuming that discordance is caused by incomplete lineage sorting, species tree estimates can be obtained by finding those species trees that minimize the number of "deep" coalescence events required for a given collection of gene trees. Efficient algorithms now exist for applying the minimizing-deep-coalescence (MDC) criterion, and simulation experiments have demonstrated its promising performance. However, it has also been noted from simulation results that the MDC criterion is not always guaranteed to infer the correct species tree estimate. In this article, we investigate the consistency of the MDC criterion. Using the multispecies coalescent model, we show that there are indeed anomaly zones for the MDC criterion for asymmetric four-taxon species tree topologies, and for all species tree topologies with five or more taxa.