We provide precise asymptotic estimates for the number of general phylogenetic networks by using analytic combinatorial methods. Recently, this approach is studied by Fuchs, Gittenberger, and the author himself (Australasian Journal of Combinatorics 73(2):385-423, 2019), to count networks with few reticulation vertices for two subclasses: tree-child and normal networks. We follow this line of research to show how to obtain results on the enumeration of general phylogenetic networks.
In recent decades, phylogenetic networks have become a standard tool in modeling evolutionary processes. Nevertheless, basic combinatorial questions about them are still largely open. For instance, even the asymptotic counting problem for the class of phylogenetic networks and subclasses is unsolved. In this paper, we propose a method based on generating functions to count networks with few reticulation vertices for two subclasses which are important in applications: tree-child networks and normal networks. In particular, our method can be used to completely solve the asymptotic counting probl…
Phylogenetic networks generalize phylogenetic trees, and have been introduced in order to describe evolution in the case of transfer of genetic material between coexisting species. There are many classes of phylogenetic networks, which can all be modeled as families of graphs with labeled leaves. In this paper, we focus on rooted and unrooted level-k networks and provide enumeration formulas (exact and asymptotic) for rooted and unrooted level-1 and level-2 phylogenetic networks with a given number of leaves. We also prove that the distribution of some parameters of these networks (such as the…
In previous work, we gave asymptotic counting results for the number of tree-child and normal networks with $k$ reticulation vertices and explicit exponential generating functions of the counting sequences for $k=1,2,3$. The purpose of this note is two-fold. First, we make some corrections to our previous approach which overcounted the above numbers and thus gives erroneous exponential generating functions (however, the overcounting does not effect our asymptotic counting results). Secondly, we use our (corrected) exponential generating functions to derive explicit formulas for the number of t…
Tree-child networks, one of the prominent network classes in phylogenetics, have been introduced for the purpose of modeling reticulate evolution. Recently, the first author together with Gittenberger and Mansouri (2019) showed that the number ${\rm TC}_{\ell,k}$ of tree-child networks with $\ell$ leaves and $k$ reticulation vertices has the first-order asymptotics \[ {\rm TC}_{\ell,k}\sim c_k\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1},\qquad (\ell\rightarrow\infty). \] Moreover, they also computed $c_k$ for $k=1,2,$ and $3$. In this short note, we give a second approach to the above resul…
The need for structures capable of accommodating complex evolutionary signals such as those found in, for example, wheat has fueled research into phylogenetic networks. Such structures generalize the standard phylogenetic tree model by also allowing cycles and have been introduced in rooted and unrooted form. In contrast to phylogenetic trees, however, surprisingly little is known about the interplay between both types thus hampering our ability to make much needed progress for rooted phylogenetic networks by drawing on insights from their much better understood unrooted counterparts. Unrooted…
Rooted triples, rooted binary phylogenetic trees on three leaves, are sufficient to encode rooted binary phylogenetic trees. That is, if $\mathcal T$ and $\mathcal T'$ are rooted binary phylogenetic $X$-trees that infers the same set of rooted triples, then $\mathcal T$ and $\mathcal T'$ 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 roote…
Rooted phylogenetic networks provide a more complete representation of the ancestral relationship between species than phylogenetic trees when reticulate evolutionary processes are at play. One way to reconstruct a phylogenetic network is to consider its `ancestral profile' (the number of paths from each ancestral vertex to each leaf). In general, this information does not uniquely determine the underlying phylogenetic network. A recent paper considered a new class of phylogenetic networks called `orchard networks' where this uniqueness was claimed to hold. Here we show that an additional rest…
We study a new invariant of circular planar electrical networks, well known to phylogeneticists: the circular split system. We use our invariant to answer some open questions about levels of complexity of networks and their related Kalmanson metrics. The key to our analysis is the realization that certain matrices arising from weighted split systems are studied in another guise: the Kron reductions of Laplacian matrices of planar electrical networks. Specifically we show that a response matrix of a circular planar electrical network corresponds to a unique resistance metric obeying the Kalmans…
Phylogenetic networks are a generalization of phylogenetic trees to leaf-labeled directed acyclic graphs that represent ancestral relationships between species whose past includes non-tree-like events such as hybridization and horizontal gene transfer. Indeed, each phylogenetic network embeds a collection of phylogenetic trees. Referring to the collection of trees that a given phylogenetic network $N$ embeds as the display set of $N$, several questions in the context of the display set of $N$ have recently been analyzed. For example, the widely studied Tree-Containment problem asks if a given …
Balanced minimum evolution is a distance-based criterion for the reconstruction of phylogenetic trees. Several algorithms exist to find the optimal tree with respect to this criterion. One approach is to minimize a certain linear functional over an appropriate polytope. Here we present polytopes that allow a similar linear programming approach to finding phylogenetic networks. We investigate a two-parameter family of polytopes that arise from phylogenetic networks, and which specialize to the Balanced Minimum Evolution polytopes as well as the Symmetric Travelling Salesman polytopes. We show t…
We describe Galois connections which arise between two kinds of combinatorial structures, both of which generalize trees with labelled leaves, and then apply those connections to a family of polytopes. The graphs we study can be imbued with metric properties or associated to vectors. Famous examples are the Billera-Holmes-Vogtmann metric space of phylogenetic trees, and the Balanced Minimal Evolution polytopes of phylogenetic trees described by Eickmeyer, Huggins, Pachter and Yoshida. Recently the space of trees has been expanded to split networks by Devadoss and Petti, while the definition of…
Unrooted phylogenetic networks are graphs used to represent evolutionary relationships. Accurately reconstructing such networks is of great relevance for evolutionary biology. It has recently been conjectured that all phylogenetic networks with at least five leaves can be uniquely reconstructed from their subnetworks obtained by deleting a single leaf and suppressing degree-2 vertices. Here, we show that this conjecture is false, by presenting a counter example for each possible number of leaves that is at least~4. Moreover, we show that the conjecture is still false when restricted to binary …
We describe a method that will reconstruct an unrooted binary phylogenetic level-1 network on n taxa from the set of all quartets containing a certain fixed taxon, in O(n^3) time. We also present a more general method which can handle more diverse quartet data, but which takes O(n^6) time. Both methods proceed by solving a certain system of linear equations over GF(2). For a general dense quartet set (containing at least one quartet on every four taxa) our O(n^6) algorithm constructs a phylogenetic level-1 network consistent with the quartet set if such a network exists and returns an (O(n^2) …
A phylogenetic network is a graph-theoretical tool that is used by biologists to represent the evolutionary history of a collection of species. One potential way of constructing such networks is via a distance-based approach, where one is asked to find a phylogenetic network that in some way represents a given distance matrix, which gives information on the evolutionary distances between present-day taxa. Here, we consider the following question. For which~$k$ are unrooted level-$k$ networks uniquely determined by their distance matrices? We consider this question for shortest distances as wel…
Rooted phylogenetic networks provide an explicit representation of the evolutionary history of a set $X$ of sampled species. In contrast to phylogenetic trees which show only speciation events, networks can also accommodate reticulate processes (for example, hybrid evolution, endosymbiosis, and lateral gene transfer). A major goal in systematic biology is to infer evolutionary relationships, and while phylogenetic trees can be uniquely determined from various simple combinatorial data on $X$, for networks the reconstruction question is much more subtle. Here we ask when can a network be unique…
Tree-child networks are an important network class which are used in phylogenetics to model reticulate evolution. In a recent paper, Pons and Batle (2021) conjectured a relation between tree-child networks and certain words. In this short note, we prove their conjecture for the (important) class of one-component tree-child networks.
Phylogenetic trees canonically arise as embeddings of phylogenetic networks. We recently showed that the problem of deciding if two phylogenetic networks embed the same sets of phylogenetic trees is computationally hard, \blue{in particular, we showed it to be $Π^P_2$-complete}. In this paper, we establish a polynomial-time algorithm for this decision problem if the initial two networks consists of a normal network and a tree-child network. The running time of the algorithm is quadratic in the size of the leaf sets.
Phylogenetic networks are rooted acyclic directed graphs in which the leaves are identified with members of a set X of species. The cluster of a vertex is the set of leaves that are descendants of the vertex. A network is "distinct-cluster" if distinct vertices have distinct clusters. This paper focuses on the set DC(X) of distinct-cluster networks whose leaves are identified with the members of X. For a fixed X, a metric on DC(X) is defined. There is a "cluster-preserving" simplification process by which vertices or certain arcs may be removed without changing the clusters of any remaining ve…
Phylogenetic networks are a type of directed acyclic graph that represent how a set $X$ of present-day species are descended from a common ancestor by processes of speciation and reticulate evolution. In the absence of reticulate evolution, such networks are simply phylogenetic (evolutionary) trees. Moreover, phylogenetic networks that are not trees can sometimes be represented as phylogenetic trees with additional directed edges placed between their edges. Such networks are called {\em tree based}, and the class of phylogenetic networks that are tree based has recently been characterised. In …
Phylogenetic networks are a generalization of phylogenetic trees that are used to represent reticulate evolution. Unrooted phylogenetic networks form a special class of such networks, which naturally generalize unrooted phylogenetic trees. In this paper we define two operations on unrooted phylogenetic networks, one of which is a generalization of the well-known nearest-neighbor interchange (NNI) operation on phylogenetic trees. We show that any unrooted phylogenetic network can be transformed into any other such network using only these operations. This generalizes the well-known fact that an…
Phylogenetic networks have gained prominence over the years due to their ability to represent complex non-treelike evolutionary events such as recombination or hybridization. Popular combinatorial objects used to construct them are triplet systems and cluster systems, the motivation being that any network $N$ induces a triplet system $\mathcal R(N)$ and a softwired cluster system $\mathcal S(N)$. Since in real-world studies it cannot be guaranteed that all triplets/softwired clusters induced by a network are available it is of particular interest to understand whether subsets of $\mathcal R(N)…
Phylogenetic networks are notoriously difficult to reconstruct. Here we suggest that it can be useful to view unknown genetic distance along edges in phylogenetic networks as analogous to unknown resistance in electric circuits. This resistance distance, well known in graph theory, turns out to have nice mathematical properties which allow the precise reconstruction of networks. Specifically we show that the resistance distance for a weighted 1-nested network is Kalmanson, and that the unique associated circular split network fully represents the splits of the original phylogenetic network (or…
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…
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,…
'Tree-based' phylogenetic networks proposed by Francis and Steel have attracted much attention of theoretical biologists in the last few years. At the heart of the definitions of tree-based phylogenetic networks is the notion of 'support trees', about which there are numerous algorithmic problems that are important for evolutionary data analysis. Recently, Hayamizu (arXiv:1811.05849 [math.CO]) proved a structure theorem for tree-based phylogenetic networks and obtained linear-time and linear-delay algorithms for many basic problems on support trees, such as counting, optimisation, and enumerat…
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 p…
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. …
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 …
Driven by the need for better models that allow one to shed light into the question how life's diversity has evolved, phylogenetic networks have now joined phylogenetic trees in the center of phylogenetics research. Like phylogenetic trees, such networks canonically induce collections of phylogenetic trees, clusters, and triplets, respectively. Thus it is not surprising that many network approaches aim to reconstruct a phylogenetic network from such collections. Related to the well-studied perfect phylogeny problem, the following question is of fundamental importance in this context: When does…
In biodiversity conservation it is often necessary to prioritize the species to conserve. Existing approaches to prioritization, e.g. the Fair Proportion Index and the Shapley Value, are based on phylogenetic trees and rank species according to their contribution to overall phylogenetic diversity. However, in many cases evolution is not treelike and thus, phylogenetic networks have come to the fore as a generalization of phylogenetic trees, allowing for the representation of non-treelike evolutionary events, such as horizontal gene transfer or hybridization. Here, we extend the concepts of phy…
Phylogenetic networks are becoming of increasing interest to evolutionary biologists due to their ability to capture complex non-treelike evolutionary processes. From a combinatorial point of view, such networks are certain types of rooted directed acyclic graphs whose leaves are labelled by, for example, species. A number of mathematically interesting classes of phylogenetic networks are known. These include the biologically relevant class of stable phylogenetic networks whose members are defined via certain "fold-up" and "un-fold" operations that link them with concepts arising within the th…
Phylogenetic networks are a generalization of phylogenetic trees allowing for the representation of non-treelike evolutionary events such as hybridization. Typically, such networks have been analyzed based on their `level', i.e. based on the complexity of their 2-edge-connected components. However, recently the question of how `treelike' a phylogenetic network is has become the center of attention in various studies. This led to the introduction of \emph{tree-based networks}, i.e. networks that can be constructed from a phylogenetic tree, called the \emph{base tree}, by adding additional edges…
In phylogenetics, it is important for the phylogenetic network model parameters to be identifiable so that the evolutionary histories of a group of species can be consistently inferred. However, as the complexity of the phylogenetic network models grows, the identifiability of network models becomes increasingly difficult to analyze. As an attempt to analyze the identifiability of network models, we check whether two networks are distinguishable. In this paper, we specifically study the distinguishability of phylogenetic network models associated with level-2 networks. Using an algebraic appro…
An important problem in evolutionary biology is to reconstruct the evolutionary history of a set $X$ of species. This history is often represented as a phylogenetic network, that is, a connected graph with leaves labelled by elements in $X$ (for example, an evolutionary tree), which is usually also binary, i.e. all vertices have degree 1 or 3. A common approach used in phylogenetics to build a phylogenetic network on $X$ involves constructing it from networks on subsets of $X$. Here we consider the question of which (unrooted) phylogenetic networks are leaf-reconstructible, i.e. which networks…
Rearrangement operations transform a phylogenetic tree into another one and hence induce a metric on the space of phylogenetic trees. Popular operations for unrooted phylogenetic trees are NNI (nearest neighbour interchange), SPR (subtree prune and regraft), and TBR (tree bisection and reconnection). Recently, these operations have been extended to unrooted phylogenetic networks, which are generalisations of phylogenetic trees that can model reticulated evolutionary relationships. Here, we study global and local properties of spaces of phylogenetic networks under these three operations. In par…
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.…
Phylogenetic networks are generalizations of phylogenetic trees that allow the representation of reticulation events such as horizontal gene transfer or hybridization, and can also represent uncertainty in inference. A subclass of these, tree-based phylogenetic networks, have been introduced to capture the extent to which reticulate evolution nevertheless broadly follows tree-like patterns. Several important operations that change a general phylogenetic network have been developed in recent years, and are important for allowing algorithms to move around spaces of networks; a vital ingredient i…
Invariants for complicated objects such as those arising in phylogenetics, whether they are invariants as matrices, polynomials, or other mathematical structures, are important tools for distinguishing and working with such objects. In this paper, we generalize a complete polynomial invariant on trees to a class of phylogenetic networks called separable networks, which will include orchard networks. Networks are becoming increasingly important for their ability to represent reticulation events, such as hybridization, in evolutionary history. We provide a function from the space of internally m…
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…
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…
Phylogenetic networks represent evolutionary history of species and can record natural reticulate evolutionary processes such as horizontal gene transfer and gene recombination. This makes phylogenetic networks a more comprehensive representation of evolutionary history compared to phylogenetic trees. Stochastic processes for generating random trees or networks are important tools in evolutionary analysis, especially in phylogeny reconstruction where they can be utilized for validation or serve as priors for Bayesian methods. However, as more network generators are developed, there is a lack o…
The mathematical analysis of random phylogenetic networks via analytic and algorithmic methods has received increasing attention in the past years. In the present work we introduce branching process methods to their study. This approach appears to be new in this context. Our main results focus on random level-$k$ networks with $n$ labelled leaves. Although the number of reticulation vertices in such networks is typically linear in $n$, we prove that their asymptotic global and local shape is tree-like in a well-defined sense. We show that the depth process of vertices in a large network conver…
Phylogenetic networks are a generalization of phylogenetic trees that allow for representation of reticulate evolution. Recently, a space of unrooted phylogenetic networks was introduced, where such a network is a connected graph in which every vertex has degree 1 or 3 and whose leaf-set is a fixed set $X$ of taxa. This space, denoted $\mathcal{N}(X)$, is defined in terms of two operations on networks -- the nearest neighbor interchange and triangle operations -- which can be used to transform any network with leaf set $X$ into any other network with that leaf set. In particular, it gives rise…
