Tree structures for proximity data

Colonius, Hans; Schulze, Hans-Henning

doi:10.1111/j.2044-8317.1981.tb00626.x

Cited by 94 publications

(56 citation statements)

References 13 publications

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“…Whenever a dissimilarity measure satisfies the four-point condition, it can be represented by an additive tree (Buneman, 1971;Colonius 6 Schulze, 1979;Cunningham, 1978). That is, there exists a tree with its vertices representing ideals and alternatives, such that the path length distances represent the dissimilarities associated with the alternatives and the ideals.…”

Section: Appendix A: Proofs Of Theoremsmentioning

confidence: 99%

A Probabilistic Feature Model for Unfolding Tested for Perfect and Imperfect Nestings

Candel¹

1997

Journal of Mathematical Psychology

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Section: Appendix A: Proofs Of Theoremsmentioning

confidence: 99%

A Probabilistic Feature Model for Unfolding Tested for Perfect and Imperfect Nestings

Candel¹

1997

Journal of Mathematical Psychology

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“…In the context of the rooted triple consistency problem, we also refer to the work of [8,7], where the conditions necessary for a given set of triple constraints to define a tree are investigated.…”

Section: Whether a Tree T Exists Such That T |{A B C} Is Homeomorphmentioning

confidence: 99%

Computing the Local Consensus of Trees

1998

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Abstract. The inference of consensus from a set of evolutionary trees is a fundamental problem in a number of fields such as biology and historical linguistics, and many models for inferring this consensus have been proposed. In this paper we present a model for deriving what we call a local consensus tree T from a set of trees T . The model we propose presumes a function f , called a total local consensus function, which determines for every triple A of species, the form that the local consensus tree should take on A. We show that all local consensus trees, when they exist, can be constructed in polynomial time and that many fundamental problems can be solved in linear time. We also consider partial local consensus functions and study optimization problems under this model. We present linear time algorithms for several variations. Finally we point out that the local consensus approach ties together many previous approaches to constructing consensus trees.Key words. algorithms, graphs, evolutionary trees AMS subject classifications. 05C05, 68Q25, 92-08, 92B05PII. S0097539795287642 1.Introduction. An evolutionary tree (also called a phylogeny or phylogenetic tree) for a species set S is a rooted tree with |S| = n leaves labeled by distinct elements in S. Because evolutionary history is difficult to determine (it is both computationally difficult as most optimization problems in this area are NP-hard and scientifically difficult as well since a range of approaches appropriate to different types of data exist), a common approach to solving this problem is to apply many different algorithms to a given data set, or to different data sets representing the same species set, and then look for common elements from the set of trees which are returned.There is extensive literature about inferring consensus from ordered sets of trees, with much attention paid to the properties of the rules for inferring the consensus. In this paper, we will make an explicit assumption that the consensus rule be independent of the ordering of the trees in the input; i.e., we will presume that the input to the consensus problem is an unordered multiset of evolutionary trees, each leaf-labelled by the elements in S. We call this input a profile, noting that in this paper the terminology is restricted in meaning as we have indicated.Several consensus methods are described in the literature for deriving one tree from a profile of evolutionary trees. These methods include maximum agreement subtrees [16,19,13,24,14], strict consensus trees [4,9], median trees (also known as majority trees) [5], compatibility trees [10,11,12], the Nelson tree [22], and the Adams consensus [1].The algorithms for some of these are implemented in standard packages and are in use; most common, perhaps, are strict and majority consensus tree approaches.

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“…phylogenetic trees in which just the combinatorial structure or topology of the tree is taken into account and edge weights are ignored, Colonius and Schulze gave a characterisation in 1977 for when a collection of unweighted quartet trees is induced by a (necessarily unique) unweighted phylogenetic tree [7,8] (see also [1] for an alternative characterisation). However, it was not until much more recently in 2003 that Dress and Erdös gave an analogous result in the weighted setting for binary phylogenetic trees (trees in which every internal vertex has degree 3) [9].…”

Section: Introductionmentioning

confidence: 99%

Encoding phylogenetic trees in terms of weighted quartets

et al. 2007

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One of the main problems in phylogenetics is to develop systematic methods for constructing evolutionary or phylogenetic trees. For a set of species X, an edge-weighted phylogenetic X-tree or phylogenetic tree is a (graph theoretical) tree with leaf set X and no degree 2 vertices, together with a map assigning a non-negative length to each edge of the tree. Within phylogenetics, several methods have been proposed for constructing such trees that work by trying to piece together quartet trees on X, i.e. phylogenetic trees each having four leaves in X. Hence, it is of interest to characterise when a collection of quartet trees corresponds to a (unique) phylogenetic tree. Recently, Dress and Erdös provided such a characterisation for binary phylogenetic trees, that is, phylogenetic trees all of whose internal vertices have degree 3. Here we provide a new characterisation for arbitrary phylogenetic trees.

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Tree structures for proximity data

Cited by 94 publications

References 13 publications

A Probabilistic Feature Model for Unfolding Tested for Perfect and Imperfect Nestings

A Probabilistic Feature Model for Unfolding Tested for Perfect and Imperfect Nestings

Computing the Local Consensus of Trees

Encoding phylogenetic trees in terms of weighted quartets

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