Agreement Forests Research Articles

Agreement forests of caterpillar trees: Complexity, kernelization and branching

Given a set X of species, a phylogenetic tree is an unrooted binary tree whose leaves are bijectively labelled by X. Such trees can be used to show the way species evolve over time. One way of understanding how topologically different two phylogenetic trees are, is to construct a minimum-size agreement forest: a partition of X into the smallest number of blocks, such that the blocks induce homeomorphic, non-overlapping subtrees in both trees. This is called a maximum agreement forest. This comparison yields insight into commonalities and differences in the evolution of X across the two trees. Computing a maximum agreement forest is NP-hard [8]. In this work we study the problem on caterpillars, which are path-like phylogenetic trees. We will demonstrate that, even if we restrict the input to this highly restricted subclass, the problem remains NP-hard and is in fact APX-hard. Furthermore we show that for caterpillars two standard reduction rules well known in the literature yield a tight kernel of size at most 7k, compared to 15k for general trees [11]. Finally we demonstrate that we can determine if two caterpillars have an agreement forest with at most k blocks in O⁎(2.49k) time, compared to O⁎(3k) for general trees [4], where O⁎(.) suppresses polynomial factors.

Espalier: Efficient Tree Reconciliation and Ancestral Recombination Graphs Reconstruction Using Maximum Agreement Forests.

In the presence of recombination individuals may inherit different regions of their genome from different ancestors, resulting in a mosaic of phylogenetic histories across their genome. Ancestral recombination graphs (ARGs) can capture how phylogenetic relationships vary across the genome due to recombination, but reconstructing ARGs from genomic sequence data is notoriously difficult. Here, we present a method for reconciling discordant phylogenetic trees and reconstructing ARGs using maximum agreement forests (MAFs). Given two discordant trees, a MAF identifies the smallest possible set of topologically concordant subtrees present in both trees. We show how discordant trees can be reconciled through their MAF in a way that retains discordances strongly supported by sequence data while eliminating conflicts likely attributable to phylogenetic noise. We further show how MAFs and our reconciliation approach can be combined to select a path of local trees across the genome that maximizes the likelihood of the genomic sequence data, minimizes discordance between neighboring local trees, and identifies the recombination events necessary to explain remaining discordances to obtain a fully connected ARG. While heuristic, our ARG reconstruction approach is often as accurate as more exact methods while being much more computationally efficient. Moreover, important demographic parameters such as recombination rates can be accurately estimated from reconstructed ARGs. Finally, we apply our approach to plant infecting RNA viruses in the genus Potyvirus to demonstrate how true recombination events can be disentangled from phylogenetic noise using our ARG reconstruction methods.

Reflections on kernelizing and computing unrooted agreement forests

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

New Reduction Rules for the Tree Bisection and Reconnection Distance

Recently it was shown that, if the subtree and chain reduction rules have been applied exhaustively to two unrooted phylogenetic trees, the reduced trees will have at most 15k-9 taxa where k is the TBR (Tree Bisection and Reconnection) distance between the two trees, and that this bound is tight. Here, we propose five new reduction rules and show that these further reduce the bound to 11k-9. The new rules combine the “unrooted generator” approach introduced in Kelk and Linz (SIAM J Discrete Math 33(3):1556–1574, 2019) with a careful analysis of agreement forests to identify (i) situations when chains of length 3 can be further shortened without reducing the TBR distance, and (ii) situations when small subtrees can be identified whose deletion is guaranteed to reduce the TBR distance by 1. To the best of our knowledge these are the first reduction rules that strictly enhance the reductive power of the subtree and chain reduction rules.

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 problems that arise in the context of computing the SNPR-distance. For a phylogenetic tree $T$ and a phylogenetic network $N$, we show how this distance can be computed by considering the set of trees that are embedded in $N$ and then use this result to characterise the SNPR-distance between $T$ and $N$ in terms of agreement forests. Furthermore, we analyse properties of shortest SNPR-sequences between two phylogenetic networks $N$ and $N'$, and answer the question whether or not any of the classes of tree-child, reticulation-visible, or tree-based networks isometrically embeds into the class of all phylogenetic networks under SNPR.

Attaching leaves and picking cherries to characterise the hybridisation number for a set of phylogenies

Throughout the last decade, we have seen much progress towards characterising and computing the minimum hybridisation number for a set P of rooted phylogenetic trees. Roughly speaking, this minimum quantifies the number of hybridisation events needed to explain a set of phylogenetic trees by simultaneously embedding them into a phylogenetic network. From a mathematical viewpoint, the notion of agreement forests is the underpinning concept for almost all results that are related to calculating the minimum hybridisation number for when |P|=2. However, despite various attempts, characterising this number in terms of agreement forests for |P|>2 remains elusive. In this paper, we characterise the minimum hybridisation number for when P is of arbitrary size and consists of not necessarily binary trees. Building on our previous work on cherry-picking sequences, we first establish a new characterisation to compute the minimum hybridisation number in the space of tree-child networks. Subsequently, we show how this characterisation extends to the space of all rooted phylogenetic networks. Moreover, we establish a particular hardness result that gives new insight into some of the limitations of agreement forests.

Extremal Distances for Subtree Transfer Operations in Binary Trees

Three standard subtree transfer operations for binary trees, used in particular for phylogenetic trees, are: tree bisection and reconnection (TBR), subtree prune and regraft (SPR), and rooted subtree prune and regraft (rSPR). We show that for a pair of leaf-labelled binary trees with n leaves, the maximum number of such moves required to transform one into the other is n-Theta (sqrt{n}), extending a result of Ding, Grünewald, and Humphries, and this holds also if one of the trees is fixed arbitrarily. If the pair is chosen uniformly at random, then the expected number of moves required is n-Theta (n^{2/3}). These results may be phrased in terms of agreement forests: we also give extensions for more than two binary trees.

Autumn Algorithm-Computation of Hybridization Networks for Realistic Phylogenetic Trees.

A minimum hybridization network is a rooted phylogenetic network that displays two given rooted phylogenetic trees using a minimum number of reticulations. Previous mathematical work on their calculation has usually assumed the input trees to be bifurcating, correctly rooted, or that they both contain the same taxa. These assumptions do not hold in biological studies and "realistic" trees have multifurcations, are difficult to root, and rarely contain the same taxa. We present a new algorithm for computing minimum hybridization networks for a given pair of "realistic" rooted phylogenetic trees. We also describe how the algorithm might be used to improve the rooting of the input trees. We introduce the concept of "autumn trees", a nice framework for the formulation of algorithms based on the mathematics of "maximum acyclic agreement forests". While the main computational problem is hard, the run-time depends mainly on how different the given input trees are. In biological studies, where the trees are reasonably similar, our parallel implementation performs well in practice. The algorithm is available in our open source program Dendroscope3, providing a platform for biologists to explore rooted phylogenetic networks. We demonstrate the utility of the algorithm using several previously studied data sets.

Fixed-Parameter and Approximation Algorithms for Maximum Agreement Forests of Multifurcating Trees

We present efficient fixed-parameter and approximation algorithms for the NP-hard problem of computing a maximum agreement forest (MAF) of a pair of multifurcating (nonbinary) rooted trees. Multifurcating trees arise naturally as a result of statistical uncertainty in current tree construction methods. The size of an MAF corresponds to the subtree prune-and-regraft distance of the two trees and is intimately connected to their hybridization number. These distance measures are essential tools for understanding reticulate evolution, such as lateral gene transfer, recombination, and hybridization. Our algorithms nearly match the running times of the currently best algorithms for the binary case. This is achieved using a combination of efficient branching rules (similar to but more complex than in the binary case) and a novel edge protection scheme that further reduces the size of the search space the algorithms need to explore.

Maximal acyclic agreement forests.

Finding the hybridization number of a pair or set of trees, [Formula: see text], is a well-studied problem in phylogenetics and is equivalent to finding a maximum acyclic agreement forest (MAAF) for [Formula: see text]. This article defines a new type of acyclic agreement forest called a maximal acyclic agreement forest (mAAF). The property for which mAAFs are "simplest" is more general and could be considered more biologically relevant than the corresponding property for MAAFs, and the set of MAAFs for any [Formula: see text] is a subset of the set of mAAFs for [Formula: see text]. This article also presents two new algorithms; one finds a mAAF for any [Formula: see text] in polynomial time and the other is an exhaustive search that finds all mAAFs for some [Formula: see text], which is also a new approach to finding the hybridization number when applied to a pair of trees. The exhaustive search algorithm is applied to a real world data set, and the findings are compared to previous results.

Fixed-Parameter Algorithms for Maximum Agreement Forests

We present new and improved fixed-parameter algorithms for computing maximum agreement forests of pairs of rooted binary phylogenetic trees. The size of such a forest for two trees corresponds to their subtree prune-and-regraft distance and, if the agreement forest is acyclic, to their hybridization number. These distance measures are essential tools for understanding reticulate evolution. Our algorithm for computing maximum acyclic agreement forests is the first depth-bounded search algorithm for this problem. Our algorithms substantially outperform the best previous algorithms for these problems.

Approximation Algorithms for Nonbinary Agreement Forests

Given two rooted phylogenetic trees on the same set of taxa $X$, the Maximum Agreement Forest (maf) problem asks to find a forest that is, in a certain sense, common to both trees and has a minimum number of components. The Maximum Acyclic Agreement Forest (maaf) problem has the additional restriction that the components of the forest cannot have conflicting ancestral relations in the input trees. There has been considerable interest in the special cases of these problems in which the input trees are required to be binary. However, in practice, phylogenetic trees are rarely binary, due to uncertainty about the precise order of speciation events. Here, we show that the general, nonbinary version of maf has a polynomial-time 4-approximation and a fixed-parameter tractable (exact) algorithm that runs in $O(4^k {\rm poly}(n))$ time, where $n=|X|$ and $k$ is the number of components of the agreement forest minus one. Moreover, we show that a $c$-approximation algorithm for nonbinary maf and a $d$-approximation...

Are community forestry principles at work in Ontario’s County, Municipal, and Conservation Authority forests?

Ontario’s County, Municipal and Conservation Authority forests have received little attention within the academic literature on community forestry in Canada. These “Agreement Forests”, as they were once called, are a product of the early 20th century and have been under local government management since the 1990s. Most are situated in Southern Ontario. In this article we investigate the extent to which community forestry principles are at work in these forests. Three principles— participatory governance, local benefits and multiple forest use—are analyzed using a composite score approach derived from survey data collected from nearly all of these forest organizations (response rate = 80%). Results indicate that most of these organizations do display attributes associated with community forestry principles, including a local governance process, public participation activities, local employment and multiple-use management. Traditional forestry employment is less strong than in similar studies of Crown land community forests; however, there is an important emphasis on non-timber activities. The article concludes that the County, Municipal and Conservation Authority forests represents a unique approach, which reflects the specific geographic and socio-economic context in which it resides.

On agreement forests

Tree rearrangement operations are widely used to measure the dissimilarity between phylogenetic trees with identical leaf sets. The tree bisection and reconnection ( tbr) distance for unrooted trees can be equivalently defined in terms of agreement forests. For both the tbr distance and the less general subtree prune and regraft ( spr) distance, we use such forests to derive new upper and lower bounds on the maximal possible distance between two trees with n leaves.

HybridNET: a tool for constructing hybridization networks

When reticulation events occur, the evolutionary history of a set of existing species can be represented by a hybridization network instead of an evolutionary tree. When studying the evolutionary history of a set of existing species, one can obtain a phylogenetic tree of the set of species with high confidence by looking at a segment of sequences or a set of genes. When looking at another segment of sequences, a different phylogenetic tree can be obtained with high confidence too. This indicates that reticulation events may occur. Thus, we have the following problem: given two rooted phylogenetic trees on a set of species that correctly represent the tree-like evolution of different parts of their genomes, what is the hybridization network with the smallest number of reticulation events to explain the evolution of the set of species under consideration? We develop a program, named HybridNet, for constructing a hybridization network with the minimum number of reticulate vertices from two input trees. We first implement the O(3(d) n)-time algorithm by Whidden et al. for computing a maximum (acyclic) agreement forest. Our program can output all the maximum (acyclic) agreement forests. We then augment the program so that it can construct an optimal hybridization network for each given maximum acyclic agreement forest. To our knowledge, this is the first time that optimal hybridization networks can be rapidly constructed. The program is available for non-commercial use, at http://www.cs.cityu.edu.hk/∼lwang/software/Hn/treeComp.html.