Basic Phylogenetic Combinatorics

Dress, Andreas; Huber, Katharina T.; Koolen, Jack H.; Moulton, Vincent; Spillner, Andreas

doi:10.1017/cbo9781139019767

Cited by 84 publications

(105 citation statements)

References 99 publications

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“…Given a rooted phylogenetic tree T and three of its leaves, there is unique triplet spanned by those leaves that is contained in T . A fundamental result in phylogenetics states that T is in fact encoded by its triplets, that is, T is the unique phylogenetic tree containing the set of triplets that arises from taking all combinations of three leaves in T [9]. This result is important since it has led to various approaches to constructing phylogenetic trees from set of triplets cf.…”

Section: Introductionmentioning

confidence: 99%

Trinets encode tree-child and level-2 phylogenetic networks

Iersel

Moulton

2013

J. Math. Biol.

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Abstract. 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 fact, Huber and Moulton recently introduced trinets as a natural extension of rooted triplets to networks. In particular, they showed that level-1 phylogenetic networks are encoded by their trinets, and also conjectured that all "recoverable" rooted phylogenetic networks are encoded by their trinets. Here we prove that recoverable binary level-2 networks and binary tree-child networks are also encoded by their trinets. To do this we prove two decomposition theorems based on trinets which hold for all recoverable binary rooted phylogenetic networks. Our results provide some additional evidence in support of the conjecture that trinets encode all recoverable rooted phylogenetic networks, and could also lead to new approaches to construct phylogenetic networks from trinets.

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Section: Introductionmentioning

confidence: 99%

Trinets encode tree-child and level-2 phylogenetic networks

Iersel

Moulton

2013

J. Math. Biol.

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“…Then A and B are said to be compatible if A ∩ B ∈ {∅, A, B}. As is well known (see, e.g., [10,28]), for any X-tree T and any two vertices v, w ∈ V (T ) the subsets L(v) and L(w) of X are compatible. Theorem 6.3.…”

Section: F ) (W) If and Only If There Exists Some U ∈V (T )−{ρmentioning

confidence: 99%

“…In many topical studies in computational biology ranging from gene onthology [9] via genome-wide association studies in population genetics [22] to evolutionary genomics [21], the following fundamental mathematical problem is encountered: Given a distance D on some set X of objects, find a dendrogram D on X (essentially a rooted tree T = (V, E) with no degree-two vertices but possibly the root whose leaf set is X together with an edge-weighting ω : E → R ≥0 ; see Figure 2 for examples) such that the distance induced by D on any two of its leaves x and y equals D (x, y). In the ideal case that the distances between any two elements of X are available, it is well understood when such a tree is uniquely determined by them, and fast algorithms for reconstructing it from them are known (see, e.g.,[10, Chapter 9.2] and [28, Chapter 7.2], where dendrograms are considered in the slightly more general forms of dated rooted X-trees and equidistant representations of dissimilarities, respectively, and [2, Chapter 3] as well as the references in all three of these sources for more on this).…”

Section: Doi 101137/130927644mentioning

confidence: 99%

“…In the ideal case that the distances between any two elements of X are available, it is well understood when such a tree is uniquely determined by them, and fast algorithms for reconstructing it from them are known (see, e.g., [10,Chapter 9.2] and [28,Chapter 7.2], where dendrograms are considered in the slightly more general forms of dated rooted X-trees and equidistant representations of dissimilarities, respectively, and [2,Chapter 3] as well as the references in all three of these sources for more on this).…”

Section: Introductionmentioning

confidence: 99%

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Distinguished Minimal Topological Lassos

Huber¹,

Kettleborough²

2015

SIAM J. Discrete Math.

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Abstract.The ease with which genomic data can now be generated using Next Generation Sequencing technologies combined with a wealth of legacy data holds great promise for exciting new insights into the evolutionary relationships between and within the kingdoms of life. At the subspecies level (e.g., varieties or strains) dendograms, that is, certain edge-weighted rooted trees whose leaves are the elements of a set X of organisms under consideration, are often used to represent those relationships. As is well known, dendrograms can be uniquely reconstructed from distances provided all distances on X are known. More often than not, real biological datasets do not satisfy this assumption, implying that the sought dendrogram need not be uniquely determined by the available distances with regard to topology, edge-weighting, or both. To better understand the structural properties a set L ⊆ X 2 has to satisfy to overcome this problem, various types of lassos have been introduced. Here, we focus on the question of when a lasso uniquely determines the topology of a dendrogram; that is, it is a topological lasso for its underlying tree. We show that any set-inclusion minimal topological lasso for such a tree T can be transformed into a structurally nice minimal topological lasso for T . Calling such a lasso a distinguished minimal topological lasso for T , we characterize it in terms of the novel concept of a cluster marker map for T . In addition, we present novel results concerning the heritability of such lassos in the context of the subtree and supertree problems.Key words. dendrogram, block graph, claw-free, topological lasso, X-tree AMS subject classifications. 05C05, 92D15 DOI. 10.1137/1309276441. Introduction. In many topical studies in computational biology ranging from gene onthology [9] via genome-wide association studies in population genetics [22] to evolutionary genomics [21], the following fundamental mathematical problem is encountered: Given a distance D on some set X of objects, find a dendrogram D on X (essentially a rooted tree T = (V, E) with no degree-two vertices but possibly the root whose leaf set is X together with an edge-weighting ω : E → R ≥0 ; see Figure 2 for examples) such that the distance induced by D on any two of its leaves x and y equals D (x, y). In the ideal case that the distances between any two elements of X are available, it is well understood when such a tree is uniquely determined by them, and fast algorithms for reconstructing it from them are known (see, e.g.,[10, Chapter 9.2] and [28, Chapter 7.2], where dendrograms are considered in the slightly more general forms of dated rooted X-trees and equidistant representations of dissimilarities, respectively, and [2, Chapter 3] as well as the references in all three of these sources for more on this).The reality, however, tends to be different in many cases in that distances between pairs of objects might be missing or are not sufficiently reliable to warrant inclusion of that distance in an analysis; see, e.g., [25,26,29] for mo...

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“…Hybridization also played an important role in the evolution of bread wheat, because the findings from [7] imply that the present-day bread wheat genome is a product of multiple rounds of hybrid speciation [7]. Hence evolution is more accurately represented by an entwined network that can represent both speciation events and reticulation events (for some books on phylogenetics see [1], [8], [9], [10], [11], [12]). …”

Section: Introductionmentioning

confidence: 99%