Binary Phylogenetic Trees Research Articles

Phylo2Vec: a vector representation for binary trees

Abstract Binary phylogenetic trees inferred from biological data are central to understanding the shared history among evolutionary units. However, inferring the placement of latent nodes in a tree is computationally expensive. State-of-the-art methods rely on carefully designed heuristics for tree search, using different data structures for easy manipulation (e.g., classes in object-oriented programming languages) and readable representation of trees (e.g., Newick-format strings). Here, we present Phylo2Vec, a parsimonious encoding for phylogenetic trees that serves as a unified approach for both manipulating and representing phylogenetic trees. Phylo2Vec maps any binary tree with n leaves to a unique integer vector of length n − 1. The advantages of Phylo2Vec are fourfold: i) fast tree sampling, (ii) compressed tree representation compared to a Newick string, iii) quick and unambiguous verification if two binary trees are identical topologically, and iv) systematic ability to traverse tree space in very large or small jumps. As a proof of concept, we use Phylo2Vec for maximum likelihood inference on five real-world datasets and show that a simple hill-climbing-based optimisation scheme can efficiently traverse the vastness of tree space from a random to an optimal tree.

Computing the Bounds of the Number of Reticulations in a Tree-Child Network That Displays a Set of Trees.

Phylogenetic network is an evolutionary model that uses a rooted directed acyclic graph (instead of a tree) to model an evolutionary history of species in which reticulate events (e.g., hybrid speciation or horizontal gene transfer) occurred. Tree-child network is a kind of phylogenetic network with structural constraints. Existing approaches for tree-child network reconstruction can be slow for large data. In this study, we present several computational approaches for bounding from below the number of reticulations in a tree-child network that displays a given set of rooted binary phylogenetic trees. In addition, we also present some theoretical results on bounding from above the number of reticulations. Through simulation, we demonstrate that the new lower bounds on the reticulation number for tree-child networks can practically be computed for large tree data. The bounds can provide estimates of reticulation for relatively large data.

Embedding phylogenetic trees in networks of low treewidth

Given a rooted, binary phylogenetic network and a rooted, binary phylogenetic tree, can the tree be embedded into the network? This problem, called \textsc{Tree Containment}, arises when validating networks constructed by phylogenetic inference methods.We present the first algorithm for (rooted) \textsc{Tree Containment} using the treewidth $t$ of the input network $N$ as parameter, showing that the problem can be solved in $2^{O(t^2)}\cdot|N|$ time and space.

Defining binary phylogenetic trees using parsimony: New bounds

Phylogenetic trees are frequently used to model evolution. Such trees are typically reconstructed from data like DNA, RNA, or protein alignments using methods based on criteria like maximum parsimony (amongst others). Maximum parsimony has been assumed to work well for data with only few state changes. Recently, some progress has been made to formally prove this assertion. For instance, it has been shown that each binary phylogenetic tree T with n≥20k leaves is uniquely defined by the set Ak(T), which consists of all characters with parsimony score k on T. In the present manuscript, we show that the statement indeed holds for all n≥4k, thus drastically lowering the lower bound for n from 20k to 4k. However, it has been known that for n≤2k and k≥3, it is not generally true that Ak(T) defines T. We improve this result by showing that the latter statement can be extended from n≤2k to n≤2k+2. So we drastically reduce the gap of values of n for which it is unknown if trees T on n taxa are defined by Ak(T) from the previous interval of [2k+1,20k−1] to the interval [2k+3,4k−1]. Moreover, we close this gap completely for the nearest neighbor interchange (NNI) neighborhood of T in the following sense: We show that as long as n≥2k+3, no tree that is one NNI move away from T (and thus very similar to T) shares the same Ak-alignment.

Building a small and informative phylogenetic supertree

We combine two fundamental optimization problems related to the construction of phylogenetic trees called maximum rooted triplets consistency and minimally resolved supertree into a new problem, which we call q-maximum rooted triplets consistency (q-MAXRTC). It takes as input a set R of rooted, binary phylogenetic trees with three leaves each and asks for a phylogenetic tree with exactly q internal nodes that contains the largest possible number of trees from R. We prove that q-MAXRTC is NP-hard to approximate within a constant, develop polynomial-time approximation algorithms for different values of q, and show experimentally that representing a phylogenetic tree by one having much fewer nodes typically does not destroy too much branching information. To demonstrate the algorithmic advantage of using trees with few internal nodes, we also propose a new algorithm for computing the rooted triplet distance that is faster than the existing algorithms when restricted to such trees.

Hypercubes and Hamilton cycles of display sets of rooted phylogenetic networks

In the context of reconstructing phylogenetic networks from a collection of phylogenetic trees, several characterisations and subsequently algorithms have been established to reconstruct a phylogenetic network that collectively embeds all trees in the input in some minimum way. For many instances however, the resulting network also embeds additional phylogenetic trees that are not part of the input. However, little is known about these inferred trees. In this paper, we explore the relationships among all phylogenetic trees that are embedded in a given phylogenetic network. First, we investigate some combinatorial properties of the collection P of all rooted binary phylogenetic trees that are embedded in a rooted binary phylogenetic network N. To this end, we associated a particular graph G, which we call rSPR graph, with the elements in P and show that, if |P|=2k, where k is the number of vertices with in-degree two in N, then G has a Hamilton cycle. Second, by exploiting rSPR graphs and properties of hypercubes, we turn to the well-studied class of rooted binary level-1 networks and give necessary and sufficient conditions for when a set of rooted binary phylogenetic trees can be embedded in a level-1 network without inferring any additional trees. Lastly, we show how these conditions translate into a polynomial-time algorithm to reconstruct such a network if it exists.

Lpnet: Reconstructing phylogenetic networks from distances using integer linear programming

AbstractNeighbor‐net is a widely used network reconstructing method that approximates pairwise distances between taxa by a circular phylogenetic network.We present Lpnet, a variant of Neighbor‐net. We first apply standard methods to construct a binary phylogenetic tree and then use integer linear programming to compute an optimal circular ordering that agrees with all tree splits.This approach achieves an improved approximation of the input distance for the clear majority of experiments that we have run for simulated and real data. We release an implementation in R that can handle up to 94 taxa and usually needs about 1 min on a standard computer for 80 taxa. For larger taxa sets, we include a top‐down heuristic which also tends to perform better than Neighbor‐net.Our Lpnet provides an alternative to Neighbor‐net and performs better in most cases. We anticipate Lpent will be useful to generate phylogenetic hypotheses.

Defining Binary Phylogenetic Trees Using Parsimony

Phylogenetic (i.e., leaf-labeled) trees play a fundamental role in evolutionary research. A typical problem is to reconstruct such trees from data like DNA alignments (whose columns are often referred to as characters), and a simple optimization criterion for such reconstructions is maximum parsimony. It is generally assumed that this criterion works well for data in which state changes are rare. In the present manuscript, we prove that each binary phylogenetic tree T with nge 20 k leaves is uniquely defined by the set A_k(T), which consists of all characters with parsimony score k on T. This can be considered as a promising first step toward showing that maximum parsimony as a tree reconstruction criterion is justified when the number of changes in the data is relatively small.

Checking Phylogenetics Decisiveness in Theory and in Practice.

Suppose we aim to build a phylogeny for a set of taxa X using information from a collection of loci, where each locus offers information for only a fraction of the taxa. The question is whether, based solely on the pattern of data availability, called a taxon coverage pattern, one can determine if the data suffices to construct a reliable phylogeny. The problem can be expressed combinatorially as follows. Let us call a taxon coverage pattern decisive if, for any binary phylogenetic tree T for X, the collection of phylogenetic trees obtained by restricting T to the subset of X covered by each locus uniquely determines T. Here we relate the problem of checking whether a taxon coverage pattern is decisive to a hypergraph coloring problem. Using this connection, we (1) show that checking decisiveness is co-NP complete; (2) obtain lower bounds on the amount of coverage needed to achieve decisiveness; (3) devise an exact algorithm for decisiveness; (4) develop problem reduction rules, and use them to obtain efficient algorithms for inputs with few loci; and (5) devise Boolean satisfiability (SAT) and integer linear programming formulations (ILP) of decisiveness that allow us to analyze data sets that arise in practice. For data sets that are not decisive, we use our SAT and ILP formulations to obtain decisive subsets of the data.

Cyclic generators and an improved linear kernel for the rooted subtree prune and regraft distance

The rooted subtree prune and regraft (rSPR) distance between two rooted binary phylogenetic trees is a well-studied measure of topological dissimilarity that is NP-hard to compute. Here we describe an improved linear kernel for the problem. In particular, we show that if the classical subtree and chain reduction rules are augmented with a modified type of chain reduction rule, the resulting trees have at most 9k−3 leaves, where k is the rSPR distance; and that this bound is tight. In comparison, the previous best-known linear kernel is of size at most 28k. To achieve this improvement we introduce cyclic generators, which can be viewed as cyclic analogues of the generators used in the phylogenetic networks literature. As a corollary to our main result we also give an improved weighted linear kernel for the minimum hybridization problem on two rooted binary phylogenetic trees.

Reflections on kernelizing and computing unrooted agreement forests

Phylogenetic trees are leaf-labelled trees used to model the evolution of species. Here we explore the practical impact of kernelization (i.e. data reduction) on the NP-hard problem of computing the TBR distance between two unrooted binary phylogenetic trees. This problem is better-known in the literature as the maximum agreement forest problem, where the goal is to partition the two trees into a minimum number of common, non-overlapping subtrees. We have implemented two well-known reduction rules, the subtree and chain reduction, and five more recent, theoretically stronger reduction rules, and compare the reduction achieved with and without the stronger rules. We find that the new rules yield smaller reduced instances and thus have clear practical added value. In many cases they also cause the TBR distance to decrease in a controlled fashion, which can further facilitate solving the problem in practice. Next, we compare the achieved reduction to the known worst-case theoretical bounds of 15k-9 and 11k-9 respectively, on the number of leaves of the two reduced trees, where k is the TBR distance, observing in both cases a far larger reduction in practice. As a by-product of our experimental framework we obtain a number of new insights into the actual computation of TBR distance. We find, for example, that very strong lower bounds on TBR distance can be obtained efficiently by randomly sampling certain carefully constructed partitions of the leaf labels, and identify instances which seem particularly challenging to solve exactly. The reduction rules have been implemented within our new solver Tubro which combines kernelization with an Integer Linear Programming (ILP) approach. Tubro also incorporates a number of additional features, such as a cluster reduction and a practical upper-bounding heuristic, and it can leverage combinatorial insights emerging from the proofs of correctness of the reduction rules to simplify the ILP.

Trinets encode orchard phylogenetic networks.

Rooted triples, rooted binary phylogenetic trees on three leaves, are sufficient to encode rooted binary phylogenetic trees. That is, if [Formula: see text] and [Formula: see text] are rooted binary phylogenetic X-trees that infer the same set of rooted triples, then [Formula: see text] and [Formula: see text] are isomorphic. However, in general, this sufficiency does not extend to rooted binary phylogenetic networks. In this paper, we show that trinets, phylogenetic network analogues of rooted triples, are sufficient to encode rooted binary orchard networks. Rooted binary orchard networks naturally generalise rooted binary tree-child networks. Moreover, we present a polynomial-time algorithm for building a rooted binary orchard network from its set of trinets. As a consequence, this algorithm affirmatively answers a previously-posed question of whether there is a polynomial-time algorithm for building a rooted binary tree-child network from the set of trinets it infers.

Toric geometry of the Cavender-Farris-Neyman model with a molecular clock

We give a combinatorial description of the toric ideal of invariants of the Cavender-Farris-Neyman model with a molecular clock (CFN-MC) on a rooted binary phylogenetic tree and prove results about the polytope associated to this toric ideal. Key results about the polyhedral structure include that the number of vertices of this polytope is a Fibonacci number, the facets of the polytope can be described using the combinatorial “cluster” structure of the underlying rooted tree, and the volume is equal to an Euler zig-zag number. The toric ideal of invariants of the CFN-MC model has a quadratic Gröbner basis with squarefree initial terms. Finally, we show that the Ehrhart polynomial of these polytopes, and therefore the Hilbert series of the ideals, depends only on the number of leaves of the underlying binary tree, and not on the topology of the tree itself. These results are analogous to classic results for the Cavender-Farris-Neyman model without a molecular clock. However, new techniques are required because the molecular clock assumption destroys the toric fiber product structure that governs group-based models without the molecular clock.

Caterpillars on three and four leaves are sufficient to reconstruct binary normal networks.

While every rooted binary phylogenetic tree is determined by its set of displayed rooted triples, such a result does not hold for an arbitrary rooted binary phylogenetic network. In particular, there exist two non-isomorphic rooted binary temporal normal networks that display the same set of rooted triples. Moreover, without any structural constraint on the rooted phylogenetic networks under consideration, similarly negative results have also been established for binets and trinets which are rooted subnetworks on two and three leaves, respectively. Hence, in general, piecing together a rooted phylogenetic network from such a set of small building blocks appears insurmountable. In contrast to these results, in this paper, we show that a rooted binary normal network is determined by its sets of displayed caterpillars (particular type of subtrees) on three and four leaves. The proof is constructive and realises a polynomial-time algorithm that takes the sets of caterpillars on three and four leaves displayed by a rooted binary normal network and, up to isomorphism, reconstructs this network.

Deciding the existence of a cherry-picking sequence is hard on two trees

Here we show that deciding whether two rooted binary phylogenetic trees on the same set of taxa permit a cherry-picking sequence, a special type of elimination order on the taxa, is NP-complete. This improves on an earlier result which proved hardness for eight or more trees. Via a known equivalence between cherry-picking sequences and temporal phylogenetic networks, our result proves that it is NP-complete to determine the existence of a temporal phylogenetic network that contains topological embeddings of both trees. The hardness result also greatly strengthens previous inapproximability results for the minimum temporal-hybridization number problem. This is the optimization version of the problem where we wish to construct a temporal phylogenetic network that topologically embeds two given rooted binary phylogenetic trees and that has a minimum number of indegree-2 nodes, which represent events such as hybridization and horizontal gene transfer. We end on a positive note, pointing out that fixed parameter tractability results in this area are likely to ensure the continued relevance of the temporal phylogenetic network model.

Minimizing the deep coalescence cost.

Metagenomic studies identify the species present in an environmental sample usually by using procedures that match molecular sequences, e.g. genes, with the species taxonomy. Here, we first formulate the problem of gene-species matching in the parsimony framework using binary phylogenetic gene and species trees under the deep coalescence cost and the assumption that each gene is paired uniquely with one species. In particular, we solve the problem in the cases when one of the trees is a caterpillar. Next, we propose a dynamic programming algorithm, which solves the problem exactly, however, its time and space complexity is exponential. Next, we generalize the problem to include non-binary trees and show the solution for caterpillar trees. We then propose time and space-efficient heuristic algorithms for solving the gene-species matching problem for any input trees. Finally, we present the results of computational experiments on simulated and empirical datasets consisting of binary tree pairs.

Combinatorial Scoring of Phylogenetic Trees and Networks Based on Homoplasy-Free Characters.

Construction of phylogenetic trees and networks for extant species from their characters represents one of the key problems in phylogenomics. While solution to this problem is not always uniquely defined and there exist multiple methods for tree/network construction, it becomes important to measure how well the constructed networks capture the given character relationship across the species. We propose a novel method for measuring the specificity of a given phylogenetic network in terms of the total number of distributions of homoplasy-free character states at the leaves that the network may impose. While for binary phylogenetic trees, this number has an exact formula and depends only on the number of leaves and character states but not on the tree topology, the situation is much more complicated for nonbinary trees or networks. Nevertheless, we develop an algorithm for combinatorial enumeration of such distributions, which is applicable for arbitrary trees and networks under some reasonable assumptions. We further extend our algorithm to a special class of characters that follow Dollo's law of irreversibility.

A parameterized algorithm for the Maximum Agreement Forest problem on multiple rooted multifurcating trees

The Maximum Agreement Forest problem has been extensively studied in phylogenetics. Most previous work is on two binary phylogenetic trees. In this paper, we study a generalized version of the problem: the Maximum Agreement Forest problem on multiple rooted multifurcating phylogenetic trees, from the perspective of parameterized algorithms. By taking advantage of a new branch-and-bound strategy, a parameterized algorithm with running time O(2.42km3n4) is presented for the problem, assuming that all polytomies in the multifurcating phylogenetic trees are hard.

On the existence of a cherry-picking sequence

Recently, the minimum number of reticulation events that is required to simultaneously embed a collection P of rooted binary phylogenetic trees into a so-called temporal network has been characterized in terms of cherry-picking sequences. Such a sequence is a particular ordering on the leaves of the trees in P. However, it is well-known that not all collections of phylogenetic trees have a cherry-picking sequence. In this paper, we show that the problem of deciding whether or not P has a cherry-picking sequence is NP-complete for when P contains at least eight rooted binary phylogenetic trees. Moreover, we use automata theory to show that the problem can be solved in polynomial time if the number of trees in P and the number of cherries in each such tree are bounded by a constant.

On the Complexity of Computing MP Distance Between Binary Phylogenetic Trees

Within the field of phylogenetics there is great interest in distance measures to quantify the dissimilarity of two trees. Recently, a new distance measure has been proposed: the Maximum Parsimony (MP) distance. This is based on the difference of the parsimony scores of a single character on both trees under consideration, and the goal is to find the character which maximizes this difference. Here we show that computation of MP distance on two binary phylogenetic trees is NP-hard. This is a highly nontrivial extension of an earlier NP-hardness proof for two multifurcating phylogenetic trees, and it is particularly relevant given the prominence of binary trees in the phylogenetics literature. As a corollary to the main hardness result we show that computation of MP distance is also hard on binary trees if the number of states available is bounded. In fact, via a different reduction we show that it is hard even if only two states are available. Finally, as a first response to this hardness we give a simple Integer Linear Program (ILP) formulation which is capable of computing the MP distance exactly for small trees (and for larger trees when only a small number of character states are available) and which is used to computationally verify several auxiliary results required by the hardness proofs.