The so-called binary perfect phylogeny with persistent characters has recently been thoroughly studied in computational biology as it is less restrictive than the well known binary perfect phylogeny. Here, we focus on the notion of (binary) persistent characters, i.e. characters that can be realized on a phylogenetic tree by at most one $0 \rightarrow 1$ transition followed by at most one $1 \rightarrow 0$ transition in the tree, and analyze these characters under different aspects. First, we illustrate the connection between persistent characters and Maximum Parsimony, where we characterize persistent characters in terms of the first phase of the famous Fitch algorithm. Afterwards we focus on the number of persistent characters for a given phylogenetic tree. We show that this number solely depends on the balance of the tree. To be precise, we develop a formula for counting the number of persistent characters for a given phylogenetic tree based on an index of tree balance, namely the Sackin index. Lastly, we consider the question of how many (carefully chosen) binary characters together with their persistence status are needed to uniquely determine a phylogenetic tree and provide an upper bound for the number of characters needed.
Phylogenetic diversity indices provide a formal way to apportion 'evolutionary heritage' across species. Two natural diversity indices are Fair Proportion (FP) and Equal Splits (ES). FP is also called 'evolutionary distinctiveness' and, for rooted trees, is identical to the Shapley Value (SV), which arises from cooperative game theory. In this paper, we investigate the extent to which FP and ES can differ, characterise tree shapes on which the indices are identical, and study the equivalence of FP and SV and its implications in more detail. We also define and investigate analogues of these ind…
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. In the current study, we propose a novel method for measuring the specificity of a given phylogenetic network in terms of the total number of distributions of character states at the leaves that the networ…
Recently, the perfect phylogeny model with persistent characters has attracted great attention in the literature. It is based on the assumption that complex traits or characters can only be gained once and lost once in the course of evolution. Here, we consider a generalization of this model, namely Dollo parsimony, that allows for multiple character losses. More precisely, we take a combinatorial perspective on the notion of Dollo-$k$ characters, i.e. traits that are gained at most once and lost precisely $k$ times throughout evolution. We first introduce an algorithm based on the notion of s…
A phylogenetic tree is an acyclic graph with distinctly labeled leaves, whose internal edges have a positive weight. Given a set of n leaves, the collection of all phylogenetic trees with this leaf set can be assembled into a metric cube complex known as phylogenetic tree space, or Billera-Holmes-Vogtmann tree space. In this largely combinatorial paper, we show that the isometry group of this space is the symmetric group on n elements. This fact is relevant to distance-based analyses of phylogenetic tree sets.
A recent paper (Manceau and Lambert, 2016) developed a novel approach for describing two well-defined notions of 'species' based on a phylogenetic tree and a phenotypic partition. In this paper, we explore some further combinatorial properties of this approach and describe an extension that allows an arbitrary number of phenotypic partitions to be combined with a phylogenetic tree for these two species notions.
In this paper we apply new geometric and combinatorial methods to the study of phylogenetic mixtures. The focus of the geometric approach is to describe the geometry of phylogenetic mixture distributions for the two state random cluster model, which is a generalization of the two state symmetric (CFN) model. In particular, we show that the set of mixture distributions forms a convex polytope and we calculate its dimension; corollaries include a simple criterion for when a mixture of branch lengths on the star tree can mimic the site pattern frequency vector of a resolved quartet tree. Furtherm…
The phylogenetic tree space introduced by Billera, Holmes, and Vogtmann (BHV tree space) is a CAT(0) continuous space that represents trees with edge weights with an intrinsic geodesic distance measure. The geodesic distance measure unique to BHV tree space is well known to be computable in polynomial time, which makes it a potentially powerful tool for optimization problems in phylogenetics and phylogenomics. Specifically, there is significant interest in comparing and combining phylogenetic trees. For example, BHV tree space has been shown to be potentially useful in tree summary and consens…
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…
Many discrete mathematics problems in phylogenetics are defined in terms of the relative labeling of pairs of leaf-labeled trees. These relative labelings are naturally formalized as tanglegrams, which have previously been an object of study in coevolutionary analysis. Although there has been considerable work on planar drawings of tanglegrams, they have not been fully explored as combinatorial objects until recently. In this paper, we describe how many discrete mathematical questions on trees "factor" through a problem on tanglegrams, and how understanding that factoring can simplify analysis…
Phylogenetic networks are an extension of phylogenetic trees which are used to represent evolutionary histories in which reticulation events (such as recombination and hybridization) have occurred. A central question for such networks is that of identifiability, which essentially asks under what circumstances can we reliably identify the phylogenetic network that gave rise to the observed data? Recently, identifiability results have appeared for networks relative to a model of sequence evolution that generalizes the standard Markov models used for phylogenetic trees. However, these results are…
We show that for any two values $α, β>0 $ for which $α+β>1$ then there is a value $N$ so that for all $n \geq N$ the following holds. For any binary phylogenetic tree $T$ on $n$ leaves there is a set of $\lfloor n^α\rfloor$ characters that capture $T$, and for which each character takes at most $\lfloor n^β\rfloor$ distinct states. Here `capture' means that $T$ is the unique perfect phylogeny for these characters. Our short proof of this combinatorial result is based on the probabilistic method.
We establish a log-supermodularity property for probability distributions on binary patterns observed at the tips of a tree that are generated under any 2--state Markov process. We illustrate the applicability of this result in phylogenetics by deriving an inequality relevant to estimating expected future phylogenetic diversity under a model of species extinction. In a further application of the log-supermodularity property, we derive a purely combinatorial inequality for the parsimony score of a binary character. The proofs of our results exploit two classical theorems in the combinatorics of…
Phylogenetic networks generalize phylogenetic trees by allowing the modelization of events of reticulate evolution. Among the different kinds of phylogenetic networks that have been proposed in the literature, the subclass of binary tree-child networks is one of the most studied ones. However, very little is known about the combinatorial structure of these networks. In this paper we address the problem of generating all possible binary tree-child networks with a given number of leaves in an efficient way via reduction/augmentation operations that extend and generalize analogous operations for …
Distance-based phylogenetic algorithms attempt to solve the NP-hard least squares phylogeny problem by mapping an arbitrary dissimilarity map representing biological data to a tree metric. The set of all dissimilarity maps is a Euclidean space properly containing the space of all tree metrics as a polyhedral fan. Outputs of distance-based tree reconstruction algorithms such as UPGMA and Neighbor-Joining are points in the maximal cones in the fan. Tree metrics with polytomies lie at the intersections of maximal cones. A phylogenetic algorithm divides the space of all dissimilarity maps into reg…
Phylogenetic inference aims to reconstruct the evolutionary relationships of different species based on genetic (or other) data. Discrete characters are a particular type of data, which contain information on how the species should be grouped together. However, it has long been known that some characters contain more information than others. For instance, a character that assigns the same state to each species groups all of them together and so provides no insight into the relationships of the species considered. At the other extreme, a character that assigns a different state to each species …
The metric space of phylogenetic trees defined by Billera, Holmes, and Vogtmann, which we refer to as BHV space, provides a natural geometric setting for describing collections of trees on the same set of taxa. However, it is sometimes necessary to analyze collections of trees on non-identical taxa sets (i.e., with different numbers of leaves), and in this context it is not evident how to apply BHV space. Davidson et al. recently approached this problem by describing a combinatorial algorithm extending tree topologies to regions in higher dimensional tree spaces, so that one can quickly comput…
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…
Phylogenetic trees describe the evolutionary history of a group of present-day species from a common ancestor. These trees are typically reconstructed from aligned DNA sequence data. In this paper we analytically address the following question: is the amount of sequence data required to accurately reconstruct a tree significantly more than the amount required to test whether or not a candidate tree was the `true' tree? By `significantly', we mean that the two quantities behave the same way as a function of the number of species being considered. We prove that, for a certain type of model, the …
This paper studies a Markov chain for phylogenetic reconstruction which uses a popular transition between tree topologies known as subtree pruning-and-regrafting (SPR). We analyze the Markov chain in the simpler setting that the generating tree consists of very short edge lengths, short enough so that each sample from the generating tree (or character in phylogenetic terminology) is likely to have only one mutation, and that there enough samples so that the data looks like the generating distribution. We prove in this setting that the Markov chain is rapidly mixing, i.e., it quickly converges …
In this paper we study group-based Markov models of evolution and their mixtures. In the algebreo-geometric setting, group-based phylogenetic tree models correspond to toric varieties, while their mixtures correspond to secant and join varieties. Determining properties of these secant and join varieties can aid both in model selection and establishing parameter identifiability. Here we explore the first natural geometric property of these varieties: their dimension. The expected projective dimension of the join variety of a set of varieties is one more than the sum of their dimensions. A join …
A nearest-neighbor-interchange (NNI) walk is a sequence of unrooted phylogenetic trees, T_0, T_1, T_2,... where each consecutive pair of trees differ by a single NNI move. We give tight bounds on the length of the shortest NNI-walks that visit all trees in an subtree-prune-and-regraft (SPR) neighborhood of a given tree. For any unrooted, binary tree, T, on n leaves, the shortest walk takes θ(n^2) additional steps than the number of trees in the SPR neighborhood. This answers Bryant's Second Combinatorial Conjecture from the Phylogenetics Challenges List, the Isaac Newton Institute, 2011, and t…
In conservation biology, phylogenetic diversity (PD) provides a way to quantify the impact of the current rapid extinction of species on the evolutionary `Tree of Life'. This approach recognises that extinction not only removes species but also the branches of the tree on which unique features shared by the extinct species arose. In this paper, we investigate three questions that are relevant to PD. The first asks how many sets of species of given size $k$ preserve the maximum possible amount of PD in a given tree. The number of such maximum PD sets can be very large, even for moderate-sized p…
Tree-child networks are a recently-described class of directed acyclic graphs that have risen to prominence in phylogenetics (the study of evolutionary trees and networks). Although these networks have a number of attractive mathematical properties, many combinatorial questions concerning them remain intractable. In this paper, we show that endowing these networks with a biologically relevant ranking structure yields mathematically tractable objects, which we term ranked tree-child networks (RTCNs). We explain how to derive exact and explicit combinatorial results concerning the enumeration an…
Quartet trees displayed by larger phylogenetic trees have long been used as inputs for species tree and supertree reconstruction. Computational constraints prevent the use of all displayed quartets in many practical problems due to the number of taxa. We introduce the notion of an Efficient Quartet System (EQS) to represent a phylogenetic tree with a subset of the quartets displayed by the tree. We show mathematically that the set of quartets obtained from a tree via an EQS contains all of the combinatorial information of the tree itself. Using performance tests on simulated datasets, we also …
Coalescent histories are combinatorial structures that describe for a given gene tree and species tree the possible lists of branches of the species tree on which the gene tree coalescences take place. Properties of the number of coalescent histories for gene trees and species trees affect a variety of probabilistic calculations in mathematical phylogenetics. Exact and asymptotic evaluations of the number of coalescent histories, however, are known only in a limited number of cases. Here we introduce a particular family of species trees, the \emph{lodgepole} species trees $(λ_n)_{n\geq 0}$, in…
Here we present a new fixed parameter tractable algorithm to compute the hybridization number r of two rooted, not necessarily binary phylogenetic trees on taxon set X in time (6^r.r!).poly(n)$, where n=|X|. The novelty of this approach is its use of terminals, which are maximal elements of a natural partial order on X, and several insights from the softwired clusters literature. This yields a surprisingly simple and practical bounded-search algorithm and offers an alternative perspective on the underlying combinatorial structure of the hybridization number problem.
We analyse a maximum-likelihood approach for combining phylogenetic trees into a larger `supertree'. This is based on a simple exponential model of phylogenetic error, which ensures that ML supertrees have a simple combinatorial description (as a median tree, minimising a weighted sum of distances to the input trees). We show that this approach to ML supertree reconstruction is statistically consistent (it converges on the true species supertree as more input trees are combined), in contrast to the widely-used MRP method, which we show can be statistically inconsistent under the exponential er…
In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. The first important outcome of our study is a clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees. Generic BHV trees on $n+1$ leaf nodes fall into $(2n-1)!!$ distinct strata,…
Generalizing matrix representations to fully heterochronous ranked tree shapes
Fred Hutchinson Cancer Research Center（弗雷德·哈钦森癌症研究中心）；Howard Hughes Medical Institute（霍华德·休斯医学研究所）；Stanford University（斯坦福大学）；Stanford School of Medicine（斯坦福医学院）；University of Washington（华盛顿大学）
Rooted and ranked binary trees are mathematical objects of great importance used to model hierarchical data and evolutionary relationships with applications in many fields including evolutionary biology and genetic epidemiology. Bayesian phylogenetic inference usually explore the posterior distribution of trees via Markov Chain Monte Carlo methods, however assessing uncertainty and summarizing distributions or samples of such trees remains challenging. While labelled phylogenetic trees have been extensively studied, relatively less literature exists for unlabelled trees which are increasingly …
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 G…
