Rooted NNI moves on tree-based phylogenetic networks

A partial order and cluster-similarity metric on rooted phylogenetic trees

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…

Planar Rooted Phylogenetic Networks

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…

Classes of Explicit Phylogenetic Networks and their Biological and Mathematical Significance

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. …

系统发育树的向量编码：通过有序叶子附着方法

Vector encoding of phylogenetic trees by ordered leaf attachment

提出有序叶子附着（OLA）方法，将二叉有根系统发育树拓扑唯一编码为整数向量，编码解码线性时间，并研究其诱导距离与NNI和SPR距离的关系。

Discriminative Measures for Comparison of Phylogenetic Trees

In this paper we introduce and study three new measures for efficient discriminative comparison of phylogenetic trees. The NNI navigation dissimilarity $d_{nav}$ counts the steps along a "combing" of the Nearest Neighbor Interchange (NNI) graph of binary hierarchies, providing an efficient approximation to the (NP-hard) NNI distance in terms of "edit length". At the same time, a closed form formula for $d_{nav}$ presents it as a weighted count of pairwise incompatibilities between clusters, lending it the character of an edge dissimilarity measure as well. A relaxation of this formula to a sim…

二元正常网络无需近亲络化可从其根部三元组重建

Binary normal networks without near reticulations can be reconstructed from their rooted triples

本文研究了二元正常网络能否仅通过其显示的根部三元组进行重建，发现除了具有特定子结构的网络外，其余均可被重建。

Exploring spaces of semi-directed phylogenetic networks

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,…

On the Complexity of Optimising Variants of Phylogenetic Diversity on 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 …

系统发育网络中的非必要弧

Non-essential arcs in phylogenetic networks

本文提出有根系统发育网络的非必要弧概念，刻画 tree-child network 中可删而不改变嵌入树集合的弧，并给出三次时间删除算法及任意网络判定的 $\Pi_2^P$ 完全性结果。

有根与无根树-子网络的特征刻画

Characterising rooted and unrooted tree-child networks

本文通过樱桃采摘序列给出有根树-子网络的新特征刻画，并将其推广到无根情形，为判定无根系统发育网络能否定向为有根树-子网络提供了新思路。

A first step towards computing all hybridization networks for two rooted binary phylogenetic trees

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…

TripNet: A Heuristic Algorithm for Constructing Rooted Phylogenetic Networks from Triplets

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.

有根系统发育网络显示集的超立方体与 Hamilton 圈

Hypercubes and Hamilton cycles of display sets of rooted phylogenetic networks

本文研究有根系统发育网络所嵌入树之间的组合关系，提出 rSPR 图并证明其 Hamilton 圈性质，进而给出 level-1 网络无额外推断树的充要条件及多项式重建算法。

有根系统发育网络的一致性距离

The agreement distance of rooted phylogenetic networks

本文提出最大一致性图来推广最大一致性森林，在有根系统发育网络上定义一致性距离度量，并证明该度量以常数因子界定SNPR等推广操作诱导的距离。

Rearrangement operations on unrooted phylogenetic 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…

树重排图允许递减的罗宾逊-福尔德斯距离路径

Tree rearrangement graphs admit paths of decreasing Robinson-Foulds distance

研究证明在NNI和SPR图中，任意两棵树可通过递减RF距离的路径连接，并展示不存在严格递减NNI路径的树示例。

The Fair Proportion is a Shapley Value on phylogenetic networks too

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 …

On cherry and pitchfork distributions of random rooted and unrooted phylogenetic trees

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…

How tree-based is my network? Proximity measures for unrooted phylogenetic networks

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…

Level-k 有根系统发育网络树宽的界

Bounds on the Treewidth of Level-k Rooted Phylogenetic Networks

本文研究 level-k 系统发育网络的树宽上界与下界，证明了 (k+3)/2 及改进的 (1/3+δ)k 上界，并基于扩展图给出 k/13 下界，从而说明常数 level 网络上许多计算问题可在多项式时间内求解。

A balance index for phylogenetic trees based on rooted quartets

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.

Computing Hybridization Networks for Multiple Rooted Binary Phylogenetic Trees by Maximum Acyclic Agreement Forests

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…

使用简约性定义二元系统发育树：新的界

Defining binary phylogenetic trees using parsimony: new bounds

本文研究基于最大简约性的二元系统发育树唯一性问题，将叶子数$n$的下界从$20k$降至$4k$，将非定义区间扩展至$n \leq 2k+2$，并完全填补了最近邻交换（NNI）邻域中的空白。

The mean value of the squared path-difference distance for rooted phylogenetic trees

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.

系统发育树上的阶依赖不相似性度量

Order-Dependent Dissimilarity Measures on Phylogenetic Trees

本文比较有根系统发育树的三种阶依赖向量表示度量与rSPR距离、杂交数等指标的关系，证明在特定排序最小化下杂交数与时间树-子杂交数分别与这些向量度量等价。

Tree-Based Unrooted Nonbinary Phylogenetic Networks

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.…

A Simple Linear-Time Algorithm for the Common Refinement of Rooted Phylogenetic Trees on a Common Leaf Set

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…

Fast computation of all maximum acyclic agreement forests for two rooted binary phylogenetic trees

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…

Bounds for phylogenetic network space metrics

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…

关于有根系统发育网络类中漏斗状定向的存在性

On the existence of funneled orientations for classes of rooted phylogenetic networks

本文修正了 Bulteau 等人关于漏斗状 tree-child 定向计算困难性的错误证明，并将困难性结果推广到漏斗状 normal 和漏斗状 reticulation-visible 等有根系统发育网络类，且结果对两种不同扎根方式均成立。

Exchangeable and Sampling Consistent Distributions on Rooted Binary Trees

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 …

A structure theorem for rooted binary phylogenetic networks and its implications for tree-based networks

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 …

Nodal distances for rooted phylogenetic trees

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 arbit…

半有向系统发育网络及其可多根化变体的特征刻画

Characterizing semi-directed phylogenetic networks and their multi-rootable variants

本文针对隐式定义的半有向及多半有向系统发育网络，提出显式数学刻画，拓展有根网络工具明确其对应知名网络类别的判定条件，为相关理论与计算分析奠定基础。

将系统发育树嵌入低树宽网络中

Embedding phylogenetic trees in networks of low treewidth

本文针对系统发育网络验证中树嵌入问题Tree Containment，提出首个以网络树宽t为参数的算法，证明该问题可在2^{O(t^2)}·|N|时空内求解。

从仿射与射影视角看系统发育多样性指数

Phylogenetic diversity indices from an affine and projective viewpoint

本文引入了基于子集（簇）和二分（分裂）的更一般系统发育多样性指数，利用仿射与射影几何的离散类比建立统一框架，为基于有根和无根系统发育树的指数提供推广、新证明及新变体。

对根放置具有鲁棒性的系统发育网络类

Classes of phylogenetic networks that are robust to root placement

研究无根系统发育网络中根放置问题，建立鲁棒可定向网络结构条件，证明2级及以下或基于树的无根网络是鲁棒可定向的，还给出鲁棒类$\mathcal C$等相关结论，揭示网络与树在鲁棒性上的关系。

Normalising phylogenetic networks

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…

一类受有根树-子网络性质启发的无根系统发育网络

A Class of Unrooted Phylogenetic Networks Inspired by the Properties of Rooted Tree-Child Networks

本文提出了一种新的无向系统发育网络类别，称为q-可切网络，以替代树-子网络，具有多项式时间识别性和计算实用性。

Computing a Relevant Set of Nonbinary Maximum Acyclic Agreement Forests

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…

Phylogenetic consensus networks: Computing a consensus of 1-nested phylogenetic networks

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…

Fighting network space: it is time for an SQL-type language to filter phylogenetic networks

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…

带分支长度的有根树的多项式编码

Polynomial encoding of rooted trees with branch lengths

研究带分支长度的有根树的差异量化问题，引入将分支长度纳入递归计算的二元多项式编码，证明其在特定条件下的性质，应用于HIV-1系统发育树，能准确分离数据集，优于先前方法。

Two Results about the Sackin and Colless Indices for Phylogenetic Trees and Their Shapes

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…

Combinatorial properties of phylogenetic diversity indices

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…

简化系统发育网络的变换

Transformations to simplify phylogenetic networks

本文形式化研究系统发育网络到同叶集树的变换及其一致性，证明 LSA 树方法满足新增物种下的一致性，而多种常用及新方法不满足。

Lattice consensus: A partial order on phylogenetic trees that induces an associatively stable consensus method

There is a long tradition of the axiomatic study of consensus methods in phylogenetics that satisfy certain desirable properties. One recently-introduced property is associative stability, which is desirable because it confers a computational advantage, in that the consensus method only needs to be computed "pairwise". In this paper, we introduce a phylogenetic consensus method that satisfies this property, in addition to being "regular". The method is based on the introduction of a partial order on the set of rooted phylogenetic trees, itself based on the notion of a hierarchy-preserving map …

Phylogenetic networks that are their own fold-ups

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…

The space of equidistant phylogenetic cactuses

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…

On the Subnet Prune and Regraft Distance

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…

可在线性时间内求解最大覆盖子树问题的有根近二叉系统发育网络

Rooted Almost-binary Phylogenetic Networks for which the Maximum Covering Subtree Problem is Solvable in Linear Time

本文研究近二叉系统发育网络，利用规范分解和栅栏连接性质刻画了$δ^\ast(N)=η^\ast(N)$的条件，提出了一类可线性时间求解最大覆盖子树问题的新网络。

Locating a Tree in a Reticulation-Visible Network in Cubic Time

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…

在树空间中跨越：针对有根树和无根树的连续系统发育推断

Leaping through tree space: continuous phylogenetic inference for rooted and unrooted trees

提出一种可微的连续树空间松弛方法，用于同时探索和推断有根与无根系统发育树，提升推断准确性并适用于数据极少的经验场景。

Unrooted non-binary tree-based phylogenetic networks

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…