Dissimilarity measures for (possibly weighted) phylogenetic trees based on the comparison of their vectors of path lengths between pairs of taxa, have been present in the systematics literature since the early seventies. But, as far as rooted phylogenetic trees goes, these vectors can only separate non-weighted binary trees, and therefore these dissimilarity measures are metrics only on this class. In this paper we overcome this problem, by splitting in a suitable way each path length between two taxa into two lengths. We prove that the resulting splitted path lengths matrices single out arbitrary rooted phylogenetic trees with nested taxa and arcs weighted in the set of positive real numbers. This allows the definition of metrics on this general class by comparing these matrices by means of metrics in spaces of real-valued $n\times n$ matrices. We conclude this paper by establishing some basic facts about the metrics for non-weighted phylogenetic trees defined in this way using $L^p$ metrics on these spaces of matrices.
The path-difference metric is one of the oldest distances for the comparison of fully resolved phylogenetic trees, but its statistical properties are still quite unknown. In this paper we compute the mean value of the square of the path-difference metric between two fully resolved rooted phylogenetic trees with $n$ leaves, under the uniform distribution. This complements previous work by Steel and Penny, who computed this mean value for fully resolved unrooted phylogenetic trees.
The problem of constructing an optimal rooted phylogenetic network from a set of rooted triplets is an NP-hard problem. In this paper, we present a heuristic algorithm called TripNet which tries to construct an optimal rooted phylogenetic network from an arbitrary set of triplets. We prove some theorems to justify the performance of the algorithm.
Recently, considerable effort has been put into developing fast algorithms to reconstruct a rooted phylogenetic network that explains two rooted phylogenetic trees and has a minimum number of hybridization vertices. With the standard approach to tackle this problem being combinatorial, the reconstructed network is rarely unique. From a biological point of view, it is therefore of importance to not only compute one network, but all possible networks. In this paper, we make a first step towards approaching this goal by presenting the first algorithm---called allMAAFs---that calculates all maximu…
Statistical phylogenetic inference methods use tree rearrangement operations to perform either hill-climbing local search or Markov chain Monte Carlo across tree topologies. The canonical class of such moves are the subtree-prune-regraft (SPR) moves that remove a subtree and reattach it somewhere else via the cut edge of the subtree. Phylogenetic trees and such moves naturally form the vertices and edges of a graph, such that tree search algorithms perform a (potentially stochastic) traversal of this SPR graph. Despite the centrality of such graphs in phylogenetic inference, rather little is k…
Phylogenetic networks generalize evolutionary trees, and are commonly used to represent evolutionary histories of species that undergo reticulate evolutionary processes such as hybridization, recombination and lateral gene transfer. Recently, there has been great interest in trying to develop methods to construct rooted phylogenetic networks from triplets, that is rooted trees on three species. However, although triplets determine or encode rooted phylogenetic trees, they do not in general encode rooted phylogenetic networks, which is a potential issue for any such method. Motivated by this fa…
The Fair Proportion of a species in a phylogenetic tree is a very simple measure that has been used to assess its value relative to the overall phylogenetic diversity represented by the tree. It has recently been proved by Fuchs and Jin to be equal to the Shapley Value of the coallitional game that sends each subset of species to its rooted Phylogenetic Diversity in the tree. We prove in this paper that this result extends to the natural translations of the Fair Proportion and the rooted Phylogenetic Diversity to rooted phylogenetic networks. We also generalize to rooted phylogenetic networks …
It is a known fact that, given two rooted binary phylogenetic trees, the concept of maximum acyclic agreement forests is sufficient to compute hybridization networks with minimum hybridization number. In this work, we demonstrate by first presenting an algorithm and then showing its correctness, that this concept is also sufficient in the case of multiple input trees. More precisely, we show that for computing minimum hybridization networks for multiple rooted binary phylogenetic trees on the same set of taxa it suffices to take only maximum acyclic agreement forests into account. Moreover, th…
Metrics on rooted phylogenetic trees are integral to a number of areas of phylogenetic analysis. Cluster-similarity metrics have recently been introduced in order to limit skew in the distribution of distances, and to ensure that trees in the neighbourhood of each other have similar hierarchies. In the present paper we introduce a new cluster-similarity metric on rooted phylogenetic tree space that has an associated local operation, allowing for easy calculation of neighbourhoods, a trait that is desirable for MCMC calculations. The metric is defined by the distance on the Hasse diagram induce…
We define a new balance index for rooted phylogenetic trees based on the symmetry of the evolutive history of every set of 4 leaves. This index makes sense for multifurcating trees and it can be computed in time linear in the number of leaves. We determine its maximum and minimum values for arbitrary and bifurcating trees, and we provide exact formulas for its expected value and variance on bifurcating trees under Ford's $α$-model and Aldous' $β$-model and on arbitrary trees under the $α$-$γ$-model.
A rooted phylogenetic network is a directed acyclic graph with a single root, whose sinks correspond to a set of species. As such networks are useful for representing the evolution of species that have undergone reticulate evolution, there has been great interest in developing the theory behind and algorithms for constructing them. However, unlike evolutionary trees, these networks can be highly non-planar, which can make them difficult to visualise and interpret. Here we investigate properties of planar rooted phylogenetic networks and algorithms for deciding whether or not rooted networks ha…
The evolutionary relationships among organisms have traditionally been represented using rooted phylogenetic trees. However, due to reticulate processes such as hybridization or lateral gene transfer, evolution cannot always be adequately represented by a phylogenetic tree, and rooted phylogenetic networks that describe such complex processes have been introduced as a generalization of rooted phylogenetic trees. In fact, estimating rooted phylogenetic networks from genomic sequence data and analyzing their structural properties is one of the most important tasks in contemporary phylogenetics. …
Evolutionary scenarios displaying reticulation events are often represented by rooted phylogenetic networks. Due to biological reasons, those events occur very rarely, and, thus, networks containing a minimum number of such events, so-called minimum hybridization networks, are of particular interest for research. Moreover, to study reticulate evolution, biologist need not only a subset but all of those networks. To achieve this goal, the less complex concept of rooted phylogenetic trees can be used as building block. Here, as a first important step, the trees are disjoint into common parts, so…
Tree shape statistics are important for investigating evolutionary mechanisms mediating phylogenetic trees. As a step towards bridging shape statistics between rooted and unrooted trees, we present a comparison study on two subtree statistics known as numbers of cherries and pitchforks for the proportional to distinguishable arrangements (PDA) and the Yule-Harding-Kingman (YHK) models. Based on recursive formulas on the joint distribution of the number of cherries and that of pitchforks, it is shown that cherry distributions are log-concave for both rooted and unrooted trees under these two mo…
Semi-directed phylogenetic networks have recently emerged as a class of phylogenetic networks sitting between rooted (directed) and unrooted (undirected) phylogenetic networks as they contain both directed as well as undirected edges. While the spaces of rooted phylogenetic networks and unrooted phylogenetic networks have been analyzed in recent years and various rearrangement moves to traverse these spaces have been introduced, the results do not immediately carry over to semi-directed phylogenetic networks. Here, we propose a simple rearrangement move for semi-directed phylogenetic networks,…
Phylogenetic Diversity (PD) is a prominent quantitative measure of the biodiversity of a collection of present-day species (taxa). This measure is based on the evolutionary distance among the species in the collection. Loosely speaking, if $\mathcal{T}$ is a rooted phylogenetic tree whose leaf set $X$ represents a set of species and whose edges have real-valued lengths (weights), then the PD score of a subset $S$ of $X$ is the sum of the weights of the edges of the minimal subtree of $\mathcal{T}$ connecting the species in $S$. In this paper, we define several natural variants of the PD score …
Attempting to recognize a tree inside a phylogenetic network is a fundamental undertaking in evolutionary analysis. In the last few years, therefore, tree-based phylogenetic networks, which are defined by a spanning tree called a subdivision tree, have attracted attention of theoretical biologists. However, the application of such networks is still not easy, due to many problems whose time complexities are not clearly understood. In this paper, we provide a general framework for solving those various old or new problems from a coherent perspective, rather than analyzing the complexity of each …
There exist several methods dealing with the reconstruction of rooted phylogenetic networks explaining different evolutionary histories given by rooted binary phylogenetic trees. In practice, however, due to insufficient information of the underlying data, phylogenetic trees are in general not completely resolved and, thus, those methods can often not be applied to biological data. In this work, we make a first important step to approach this goal by presenting the first algorithm --- called allMulMAAFs --- that enables the computation of all relevant nonbinary maximum acyclic agreement forest…
Stockholm University（斯德哥尔摩大学）；Leipzig University（莱比锡大学）；University of Vienna（维也纳大学）；Max Planck Institute for Mathematics in the Sciences（马普科学数学研究所）；Universidad Nacional de Colombia（哥伦比亚国立大学）；Santa Fe Institute（圣塔菲研究所）
Rooted phylogenetic networks provide a way to describe species' relationships when evolution departs from the simple model of a tree. However, networks inferred from genomic data can be highly tangled, making it difficult to discern the main reticulation signals present. In this paper, we describe a natural way to transform any rooted phylogenetic network into a simpler canonical network, which has desirable mathematical and computational properties, and is based only on the 'visible' nodes in the original network. The method has been implemented and we demonstrate its application to some exam…
Background. The supertree problem, i.e., the task of finding a common refinement of a set of rooted trees is an important topic in mathematical phylogenetics. The special case of a common leaf set $L$ is known to be solvable in linear time. Existing approaches refine one input tree using information of the others and then test whether the results are isomorphic. Results. A linear-time algorithm, LinCR, for constructing the common refinement $T$ of $k$ input trees with a common leaf set is proposed that explicitly computes the parent function of $T$ in a bottom-up approach. Conclusion. LinCR is…
An important and well-studied problem in phylogenetics is to compute a \emph{consensus tree} so as to summarize the common features within a collection of rooted phylogenetic trees, all whose leaf-sets are bijectively labeled by the same set~(X) of species. More recently, however, it has become of interest to find a consensus for a collection of more general, rooted directed acyclic graphs all of whose sink-sets are bijectively labeled by~(X), so called rooted \emph{phylogenetic networks}. These networks are used to analyse the evolution of species that cross with one another, such as plants a…
The search space of rooted phylogenetic trees is vast and a major research focus of recent decades has been the development of algorithms to effectively navigate this space. However this space is tiny when compared with the space of rooted phylogenetic networks, and navigating this enlarged space remains a poorly understood problem. This, and the difficulty of biologically interpreting such networks, obstructs adoption of networks as tools for modelling reticulation. Here, we argue that the superimposition of biologically motivated constraints, via an SQL-style language, can both stimulate use…
The Sackin and Colless indices are two widely-used metrics for measuring the balance of trees and for testing evolutionary models in phylogenetics. This short paper contributes two results about the Sackin and Colless indices of trees. One result is the asymptotic analysis of the expected Sackin and Colless indices of a tree shape (which are full binary rooted unlabelled trees) under the uniform model where tree shapes are sampled with equal probability. Another is a short elementary proof of the closed formula for the expected Sackin index of phylogenetic trees (which are full binary rooted t…
In this work, we answer an open problem in the study of phylogenetic networks. Phylogenetic trees are rooted binary trees in which all edges are directed away from the root, whereas phylogenetic networks are rooted acyclic digraphs. For the purpose of evolutionary model validation, biologists often want to know whether or not a phylogenetic tree is contained in a phylogenetic network. The tree containment problem is NP-complete even for very restricted classes of networks such as tree-sibling phylogenetic networks. We prove that this problem is solvable in cubic time for stable phylogenetic ne…
Tree-based networks are a class of phylogenetic networks that attempt to formally capture what is meant by "tree-like" evolution. A given non-tree-based phylogenetic network, however, might appear to be very close to being tree-based, or very far. In this paper, we formalise the notion of proximity to tree-based for unrooted phylogenetic networks, with a range of proximity measures. These measures also provide characterisations of tree-based networks. One measure in particular, related to the nearest neighbour interchange operation, allows us to define the notion of "tree-based rank". This pro…
We introduce a notion of finite sampling consistency for phylogenetic trees and show that the set of finitely sampling consistent and exchangeable distributions on n leaf phylogenetic trees is a polytope. We use this polytope to show that the set of all exchangeable and infinite sampling consistent distributions on 4 leaf phylogenetic trees is exactly Aldous' beta-splitting model and give a description of some of the vertices for the polytope of distributions on 5 leaves. We also introduce a new semialgebraic set of exchangeable and sampling consistent models we call the multinomial model and …
We introduce and investigate the space of \emph{equidistant} $X$-\emph{cactuses}. These are rooted, arc weighted, phylogenetic networks with leaf set $X$, where $X$ is a finite set of species, and all leaves have the same distance from the root. The space contains as a subset the space of ultrametric trees on $X$ that was introduced by Gavryushkin and Drummond. We show that equidistant-cactus space is a CAT(0)-metric space which implies, for example, that there are unique geodesic paths between points. As a key step to proving this, we present a combinatorial result concerning \emph{ranked} ro…
In phylogenetics, phylogenetic trees are rooted binary trees, whereas phylogenetic networks are rooted arbitrary acyclic digraphs. Edges are directed away from the root and leaves are uniquely labeled with taxa in phylogenetic networks. For the purpose of validating evolutionary models, biologists check whether or not a phylogenetic tree is contained in a phylogenetic network on the same taxa. This tree containment problem is known to be NP-complete. A phylogenetic network is reticulation-visible if every reticulation node separates the root of the network from some leaves. We answer an open p…
Phylogenetic networks are rooted directed acyclic graphs that represent evolutionary relationships between species whose past includes reticulation events such as hybridisation and horizontal gene transfer. To search the space of phylogenetic networks, the popular tree rearrangement operation rooted subtree prune and regraft (rSPR) was recently generalised to phylogenetic networks. This new operation - called subnet prune and regraft (SNPR) - induces a metric on the space of all phylogenetic networks as well as on several widely-used network classes. In this paper, we investigate several probl…
A Class of Unrooted Phylogenetic Networks Inspired by the Properties of Rooted Tree-Child Networks
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特理工大学应用数学研究所）；Department of Computer Science, Middlesex University（中伦敦大学计算机科学系）；School of Computer Science, University of Auckland（奥克兰大学计算机科学学院）；Faculty of Computer Science, Dalhousie University（达尔豪斯大学计算机科学学院）
An evolutionary tree (phylogenetic tree) is a binary, rooted, unordered tree that models the evolutionary history of currently living species in which leaves are labeled by species. In this paper, we investigate the problem of finding the maximum consensus evolutionary tree from a set of given rooted triplets. A rooted triplet is a phylogenetic tree on three leaves and shows the evolutionary relationship of the corresponding three species. The mentioned problem is known to be APX-hard. We present two new heuristic algorithms. For a given set of m triplets on n species, the FastTree algorithm r…
Phylogenetic networks are a generalisation of phylogenetic trees that allow for more complex evolutionary histories that include hybridisation-like processes. It is of considerable interest whether a network can be considered `tree-like' or not, which lead to the introduction of \textit{tree-based} networks in the rooted, binary context. Tree-based networks are those networks which can be constructed by adding additional edges into a given phylogenetic tree, called the \textit{base tree}. Previous extensions have considered extending to the binary, unrooted case and the nonbinary, rooted case.…
Identifiability of the discrete tree parameter is a key property for phylogenetic models since it is necessary for statistically consistent estimation of the tree from sequence data. Algebraic methods have proven to be very effective at showing that tree and network parameters of phylogenetic models are identifiable, especially when the underlying models are group-based. However, since group-based models are time-reversible, only the unrooted tree topology is identifiable and the location of the root is not. In this note we show that the rooted tree parameter of the Cavender-Farris-Neyman Mode…
The path-difference metric is one of the oldest and most popular distances for the comparison of phylogenetic trees, but its statistical properties are still quite unknown. In this paper we compute the expected value under the Yule model of evolution of its square on the space of fully resolved rooted phylogenetic trees with n leaves. This complements previous work by Steel-Penny and Mir-Rosselló, who computed this mean value for fully resolved unrooted and rooted phylogenetic trees, respectively, under the uniform distribution.
It has recently been shown that the NP-hard problem of calculating the minimum number of hybridization events that is needed to explain a set of rooted binary phylogenetic trees by means of a hybridization network is fixed-parameter tractable if an instance of the problem consists of precisely two such trees. In this paper, we show that this problem remains fixed-parameter tractable for an arbitrarily large set of rooted binary phylogenetic trees. In particular, we present a quadratic kernel.
Rooted phylogenetic networks are rooted acyclic digraphs. They are used to model complex evolution where hybridization, recombination and other reticulation events play important roles. A rigorous definition of network compression is introduced on the basis of the recent studies of the relationships between cluster, tree and rooted phylogenetic network. The concept reveals another interesting connection between the two well-studied network classes|tree-child networks and reticulation-visible networks|and enables us to define a new class of networks for which the cluster containment problem has…
Rooted acyclic graphs appear naturally when the phylogenetic relationship of a set $X$ of taxa involves not only speciations but also recombination, horizontal transfer, or hybridization, that cannot be captured by trees. A variety of classes of such networks have been discussed in the literature, including phylogenetic, level-1, tree-child, tree-based, galled tree, regular, or normal networks as models of different types of evolutionary processes. Clusters arise in models of phylogeny as the sets $\mathtt{C}(v)$ of descendant taxa of a vertex $v$. The clustering system $\mathscr{C}_N$ compris…
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 res…
Most phylogenetic models assume that the evolutionary process is stationary and reversible. As a result, the root of the tree cannot be inferred as part of the analysis because the likelihood of the data does not depend on the position of the root. Yet defining the root of a phylogenetic tree is a key component of phylogenetic inference because it provides a point of reference for polarising ancestor/descendant relationships and therefore interpreting the tree. In this paper we investigate the effect of relaxing the reversibility assumption and allowing the position of the root to be another u…
Rooted phylogenetic networks are used to describe evolutionary histories that contain non-treelike evolutionary events such as hybridization and horizontal gene transfer. In some cases, such histories can be described by a phylogenetic base-tree with additional linking arcs, which can for example represent gene transfer events. Such phylogenetic networks are called tree-based. Here, we consider two possible generalizations of this concept to nonbinary networks, which we call tree-based and strictly-tree-based nonbinary phylogenetic networks. We give simple graph-theoretic characterizations of …
Phylogenetic trees are a central tool in understanding evolution. They are typically inferred from sequence data, and capture evolutionary relationships through time. It is essential to be able to compare trees from different data sources (e.g. several genes from the same organisms) and different inference methods. We propose a new metric for robust, quantitative comparison of rooted, labeled trees. It enables clear visualizations of tree space, gives meaningful comparisons between trees, and can detect distinct islands of tree topologies in posterior distributions of trees. This makes it poss…
