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Open Access Correspondence

On Hill et al's conjecture for calculating the subtree prune and regraft distance between phylogenies

Simone Linz

Author Affiliations

Department of Computer Science, Technical University of Catalonia, Barcelona, Spain

BMC Evolutionary Biology 2010, 10:334  doi:10.1186/1471-2148-10-334


The electronic version of this article is the complete one and can be found online at: http://www.biomedcentral.com/1471-2148/10/334


Received:21 June 2010
Accepted:29 October 2010
Published:29 October 2010

© 2010 Linz; licensee BioMed Central Ltd.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

Recently, Hill et al. [1] implemented a new software package--called SPRIT--which aims at calculating the minimum number of horizontal gene transfer events that is needed to simultaneously explain the evolution of two rooted binary phylogenetic trees on the same set of taxa. To this end, SPRIT computes the closely related so-called rooted subtree prune and regraft distance between two phylogenies. However, calculating this distance is an NP-hard problem and exact algorithms are often only applicable to small- or medium-sized problem instances. Trying to overcome this problem, Hill et al. propose a divide-and-conquer approach to speed up their algorithm and conjecture that this approach can be used to compute the rooted subtree prune and regraft distance exactly.

Results

In this note, we present a counterexample to Hill et al's conjecture and subsequently show that a modified version of their conjecture holds.

Conclusion

While Hill et al's conjecture may result in an overestimate of the rooted subtree prune and regraft distance, a slightly more restricted version of their approach gives the desired outcome and can be applied to speed up the exact calculation of this distance between two phylogenies.

Background

In recent years, one of the main research foci in the development of theoretical frameworks that aim at approaching questions in evolutionary biology turns from the reconstruction of phylogenetic trees towards the reconstruction of phylogenetic networks. This has partly been triggered by the exponentially growing amount of available sequence data arising from whole genome sequencing projects and a successive detection of genes whose sequences are chimeras of distinct ancestral gene sequences, and hence, are likely to be the result of reticulation (e.g. horizontal gene transfer or hybridization). Although evolutionary biologists are now mostly acknowledging the existence of species arising from reticulation within certain groups of organisms, the extent to which such events have influenced the evolutionary history for a set of present-day species remains controversially discussed until today. To shed light on this question, Hill et al. [1] recently published a study that is centered around the identification and quantification of horizontal gene transfer. The authors have implemented a new software package--called SPRIT--consisting of a heuristic as well as an exact algorithm, applied it to several data sets of variable size, and compared their results and running times with those obtained from other algorithms that have previously been developed to analyze reticulate evolution.

Algorithmically, SPRIT draws on ideas that are borrowed from work that has been done in the context of the graph-theoretic operation of rooted subtree prune and regraft (rSPR) which is a popular tool to quantify the dissimilarity between two trees. Loosely speaking, an rSPR operation cuts (prunes) a subtree and reattaches (regrafts) it to another part of the tree. A lower bound on the number of reticulation events that is needed to simultaneously explain two phylogenies is the minimum number of rSPR operations that transform one phylogeny into the other [2,3]. This minimum number, which is computed by SPRIT, is referred to as the rSPR distance. However, since the task of calculating this distance is an NP-hard optimization problem, the application of exact algorithms is often restricted to medium-sized data sets.

In trying to overcome this obstacle, thus to speed up SPRIT, Hill et al. propose a divide-and-conquer-type reduction that breaks the problem into several smaller and more tractable subproblems before calculating the rSPR distance for each subproblem separately. Briefly, the authors conjecture that the sum of rSPR distances over all smaller subproblems is equal to the rSPR distance of the original unreduced trees. In this note, we give a counterexample to their conjecture. Nevertheless, we subsequently show that a slightly more restricted version of their conjecture holds and can be used to exactly calculate the rSPR distance between two phylogenies by breaking the problem into smaller subproblems.

The remainder of this paper is organized as follows. The next section contains some mathematical preliminaries that are needed to formally state Hill at al's conjecture. This conjecture is then given in the subsequent section which also contains the aforementioned counterexample. We then show that a modified version of the conjecture holds in the following section. We end this note with a brief conclusion.

Preliminaries

In this section, we give some preliminary definitions that are used throughout this paper. Unless otherwise stated, the notation and terminology follows [4].

Phylogenetic Trees

A rooted binary phylogenetic X-tree <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> is a rooted tree whose root has degree two while all other interior vertices have degree three and whose leaf set is X . The set X is the label set of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and is frequently denoted by <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M64">View MathML</a>. Furthermore, let X′ be a subset of X. The minimal rooted subtree of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> that connects all the leaves in X′ is denoted by <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a>(X′) while the restriction of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a>to X′, denoted by <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a>|X′, is the rooted binary phylogenetic X′-tree obtained from <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a>(X′) by contracting all degree-two vertices apart from the root.

Rooted Subtree Prune and Regraft

Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> be a rooted binary phylogenetic X-trees. For the purposes of the upcoming definition, we view the root of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> as a vertex ρ adjoined to the original root by a pendant edge. Now, let e = {u, v} be any edge of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> that is not incident with ρ such that u is the vertex on the path from ρ to <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M65">View MathML</a>. Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> be the rooted binary phylogenetic X-tree obtained from <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> by deleting e and reattaching the resulting subtree with root v via a new edge, say f , as follows. Subdivide an edge of the component that contains ρ with a new vertex u′, join u′ and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M65">View MathML</a> with f , and contract u. Then <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> has been obtained from <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> by a rooted subtree prune and regraft (rSPR) operation. The rSPR distance between two rooted binary phylogenetic X-trees <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> is the minimum number of rSPR operations that transform <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> into <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. We denote this distance by <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M3">View MathML</a>.

Agreement Forests

Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> be two rooted binary phylogenetic X-trees. Again, to make the following work, regard the roots of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> as a vertex ρ adjoined to the original root by a pendant edge. An agreement forest <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M66">View MathML</a> for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> is a partition of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M67">View MathML</a> such that <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M87">View MathML</a> and the following properties are satisfied:

(i) for all i ∈ {ρ, 1, ..., k}, we have <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M4">View MathML</a>, and

(ii) the trees in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M68">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M69">View MathML</a> are vertex-disjoint subtrees of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>, respectively.

Throughout the remainder of this note, we will interchangeably refer to <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M5">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M70">View MathML</a> as an agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. A maximum-agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> is an agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> with the smallest number of elements over all agreement forests for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. Note that a maximum-agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> is not necessarily unique.

Bordewich and Semple [5] established the following characterization which directly relates the rSPR distance to the number of elements in a maximum-agreement forest and is crucial to many algorithms that exactly compute the rSPR distance between two rooted binary phylogenetic trees.

Theorem 1. Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a>and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>be two rooted binary phylogenetic X-trees, and let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M6">View MathML</a>be a maximum-agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a>and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. Then

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M7">View MathML</a>

Clusters

Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> be a rooted binary phylogenetic X-tree, and let A be a subset of X with |A| ≥ 2. We say that A is a cluster of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> if there is a vertex <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M65">View MathML</a> in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> whose set of descendants is precisely A. We denote this cluster by <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M8">View MathML</a>.

We next consider several different types of clusters that will play an important role in the remainder of this paper. Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> be two rooted binary phylogenetic X-trees, and let A be a cluster that is common to <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>; that is there exists a vertex <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M65">View MathML</a> in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and a vertex <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M71">View MathML</a> in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> such that <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M9">View MathML</a>. Furthermore, let u (resp. u′) be the parent vertex of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M65">View MathML</a> (resp. <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M71">View MathML</a>) in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> (resp. <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>), and let w (resp. w′) be the child vertex of u (resp. u′) with <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M72">View MathML</a> (resp. <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M73">View MathML</a>). If no proper subset of A is a common cluster of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>, we refer to A as a minimal cluster. Moreover, A is a solvable cluster if A is minimal and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M10">View MathML</a>. Lastly, we say that A is a subtree-like cluster if A is a solvable cluster and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M11">View MathML</a>. Roughly speaking, the condition <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M11">View MathML</a> is satisfied if the subtree with root w in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> is identical to the subtree with root w′ in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. We refer to <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M12">View MathML</a> as the common subtree associated with A and note that it can exclusively consist of an isolated vertex. For example, A = {1, 2, ..., 6} is a solvable cluster of the two rooted binary phylogenetic X-trees <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> that are shown in Figure 1 since <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M10">View MathML</a> = {1, 2, ..., 12}. However, as <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M13">View MathML</a>, it follows that A is not a subtree-like cluster of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>.

thumbnailFigure 1. Two rooted binary phylogenetic X-trees <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. Note that <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> have an additional vertex ρ adjoined to the original root by a pendant edge.

Now, let Θ ∈ {minimal, solvable, subtree-like}. We next describe algorithmically how to obtain a sequence of tree pairs--which is important to mathematically state Hill et al's conjecture--by decomposing two rooted binary phylogenetic X-trees <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> into smaller subtrees. As previously, view the roots of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> as a vertex ρ adjoined to the original root by a pendant edge, and regard ρ as part of the label set; that is <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M74">View MathML</a>. Setting i to be 1, let Ai be a common Θ cluster of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> with <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M14">View MathML</a>. Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M15">View MathML</a> denote the rooted binary phylogenetic tree <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M16">View MathML</a> (viewing the root of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M15">View MathML</a> as a vertex ρi adjoined to the original root by a pendant edge) and reset <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> to be the tree obtained from <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> by replacing <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M17">View MathML</a> with a new vertex ai . Analogously, let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M18">View MathML</a> denote the rooted binary phylogenetic tree <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M19">View MathML</a> (viewing the root of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M18">View MathML</a> as a vertex ρi adjoined to the original root by a pendant edge) and reset <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> to be the tree obtained from <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> by replacing <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M20">View MathML</a> with a new vertex ai . If <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> contain a Θ cluster Ai+1 with <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M21">View MathML</a>, stop or increment i by 1 and repeat this process; otherwise, stop. Eventually, we obtain a sequence

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M22">View MathML</a>

of pairs of rooted binary phylogenetic trees, where <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M23">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M24">View MathML</a> denote the two trees after the replacement of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M25">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M26">View MathML</a> with a vertex at. We call this sequence a cluster sequence of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> with respect to a specific cluster type Θ. An example of a cluster sequence with respect to Θ = solvable for the two rooted binary phylogenetic trees depicted in Figure 1 is shown in Figure 2.

thumbnailFigure 2. A cluster sequence with respect to Θ = solvable for the two rooted binary phylogenetic X-trees <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> shown in Figure 1. Details on how the tree pairs have been obtained are given in the text.

Hill et al's Conjecture and a Counterexample

We begin this section by formally stating Hill et al's conjecture which was introduced in [1].

Conjecture 2. Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a>and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>be two rooted binary phylogenetic X-trees. Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M27">View MathML</a>be a cluster sequence for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a>and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>with respect to Θ = solvable. Then

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M28">View MathML</a>

(1)

Next, we detail a counterexample to the above conjecture which is based on the two rooted binary phylogenetic X-trees <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> that are shown in Figure 1. A maximum-agreement forest <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M81">View MathML</a> for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> contains 5 elements and is shown in the top of Figure 3. By Theorem 1, this implies that <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M29">View MathML</a>. Now, consider the cluster sequence with respect to Θ = solvable for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> that contains three tree pairs and is depicted in Figure 2. The first tree pair (<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M30">View MathML</a>) consists of the restricted subtrees of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> whose leaf set is the solvable cluster A1 = {1, 2, ..., 6} of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>; thus <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M31">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M32">View MathML</a>. Similarly, the second tree pair (<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M33">View MathML</a>) consists of the restricted subtrees of the two trees that have been obtained from <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> by replacing <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M34">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M35">View MathML</a>, respectively, with a single leaf a1 whose leaf set is the solvable cluster A2 = {7, 8, ..., 12}. Lastly, the third tree pair (<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M36">View MathML</a>) can be regarded as being obtained from <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> by replacing <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M34">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M35">View MathML</a> with a leaf a1 and replacing <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M37">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M38">View MathML</a> with a leaf a2. For each tree pair (<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M39">View MathML</a>) of the cluster sequence shown in Figure 2, a maximum-agreement forest <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M75">View MathML</a> with i ∈ {1, 2, ρ} is depicted in the bottom part of Figure 3. Note that each forest <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M75">View MathML</a> is the unique maximum-agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M15">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M18">View MathML</a> Now, by Equation 1, we have

thumbnailFigure 3. Maximum-agreement forests. Top: A maximum-agreement forest <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M81">View MathML</a> for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> depicted in Figure 1. Bottom: A maximum-agreement forest <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M75">View MathML</a> for each tree pair <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M15">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M18">View MathML</a> shown in Figure 2.

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M40">View MathML</a>

which is strictly greater than <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M3">View MathML</a>; thus showing that Conjecture 2 does not hold.

Using Subtree-Like Clusters to Prove Hill et al's Conjecture

In this section, we show that Conjecture 2 holds, if we consider a subtree-like cluster instead of a solvable cluster in each iteration of computing a cluster sequence for two rooted binary phylogenetic trees. We first prove the result for a cluster sequence of size two and then see that this result generalizes to cluster sequences of greater size.

Lemma 3. Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a>and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>be two rooted binary phylogenetic X-trees. Let (<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M30">View MathML</a>), (<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M36">View MathML</a>) be a cluster sequence for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a>and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>with respect to Θ = subtree-like. Then

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M41">View MathML</a>

Proof. Let A1 be the subtree-like cluster <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M42">View MathML</a> of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. We start by making an observation that is crucial for what follows. By the definition of a subtree-like cluster, there exists a common subtree, say <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M43">View MathML</a>, that is associated with A1 in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. Clearly, <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M43">View MathML</a> is also a common subtree of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M23">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M24">View MathML</a>. Furthermore, as <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M23">View MathML</a> has been obtained from <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> by replacing <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M34">View MathML</a> with a single vertex a1 and as <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M24">View MathML</a> has been obtained from <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> by replacing <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M35">View MathML</a> with a single vertex a1, it is easily checked that <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a>|(<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M47">View MathML</a>) is a common subtree of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M23">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M24">View MathML</a>.

We now show that

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M44">View MathML</a>

(2)

Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M85">View MathML</a> be a maximum-agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M45">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M46">View MathML</a>, and let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M86">View MathML</a> be a maximum-agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M23">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M24">View MathML</a>. By the observation prior to this paragraph, it follows from Proposition 3.2 of [5] that <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M47">View MathML</a> is a subset of an element, say <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M48">View MathML</a>, in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M86">View MathML</a>. Furthermore, let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M49">View MathML</a> be the label set of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M85">View MathML</a> with ρ1 <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M49">View MathML</a>. As <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M85">View MathML</a> is an agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M45">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M46">View MathML</a> and as <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M86">View MathML</a> is such a forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M23">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M24">View MathML</a>, it follows that

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M50">View MathML</a>

is an agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. As <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M48">View MathML</a> - {a1} always contains an element, note that <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M51">View MathML</a> is never the empty set. Thus <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M84">View MathML</a> and, by Theorem 1, we have

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M52">View MathML</a>

This establishes Equation 2.

We now turn to the second part of this proof and show that

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M53">View MathML</a>

(3)

Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M81">View MathML</a> be a maximum-agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. The remainder of this part splits into two cases. First, assume that there exists an element in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M81">View MathML</a>, say <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M76">View MathML</a>, such that <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M77">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M78">View MathML</a>. Note that <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M76">View MathML</a> is the only label set with the described properties, as otherwise, <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M81">View MathML</a> is not an agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M79">View MathML</a>, and let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M80">View MathML</a>. Since <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M81">View MathML</a> is an agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>,

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M54">View MathML</a>

is such a forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M45">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M46">View MathML</a> and

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M55">View MathML</a>

is an agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M23">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M24">View MathML</a>. Second, assume that no such element <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M76">View MathML</a> exists. Hence, every element <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M82">View MathML</a> in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M81">View MathML</a> is either a subset of A1 or a subset of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M83">View MathML</a>. Furthermore, as A1 is a subtree-like cluster of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> whose associated common subtree is <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M43">View MathML</a>, it again follows from Proposition 3.2 of [5], that <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M56">View MathML</a> is a subset of an element, say <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M57">View MathML</a>, in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M81">View MathML</a>. Now, as <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M81">View MathML</a> is an agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>, it follows that

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M58">View MathML</a>

is an agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M45">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M46">View MathML</a> and

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M59">View MathML</a>

is such a forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M23">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M24">View MathML</a>. Regardless of whether or not <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M76">View MathML</a> exists, we have <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M84">View MathML</a>, and therefore,

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M60">View MathML</a>

This establishes Equation 3, and combining Equations 2 and 3 completes the proof of this lemma.

The next theorem directly follows from repeated applications of Lemma 3.

Theorem 4. Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a>and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>be two rooted binary phylogenetic X-trees. Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M27">View MathML</a> be a cluster sequence for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a>and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>with respect to Θ = subtree-like. Then

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M61">View MathML</a>

Conclusion

In this paper, we have shown that Hill et al's conjecture [1] and the underlying divide-and-conquer approach cannot be used to calculate the rSPR distance between two phylogenies exactly. To provide some intuition why this conjecture fails, consider the following. Let <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M27">View MathML</a> be a cluster sequence with respect to Θ = solvable for two rooted binary phylogenetic trees <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. Calculating a maximum-agreement forest for each tree pair (<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M39">View MathML</a>), taking their union, and, for each i ∈; {1, 2, ..., t}, joining the element containing ai with the element containing ρi can potentially result in a set, say <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M62">View MathML</a>, which contains an element that is a subset of {a1, a2, ..., at , ρ1, ρ2, ..., ρt}. In the case of our counterexample,

<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M63">View MathML</a>

contains one such element. Trivially, this element is not part of any agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> while <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M62">View MathML</a> - {{a1, a2, ρ1, ρ2}} is precisely a maximum-agreement forest for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. Consequently, a divide-and-conquer approach that exactly calculates <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M3">View MathML</a> needs to take into account the number of elements in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M62">View MathML</a> that are subsets of {a1, a2, ..., at , ρ1, ρ2, ..., ρt}; otherwise, the result may be an overestimate of the exact solution. Alternatively, one can approach the problem by finding a strategy which guarantees that no element in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M62">View MathML</a> is a subset of {a1, a2, ..., at , ρ1, ρ2, ..., ρt }. This is the underlying idea of Theorem 4 which uses a slightly more restricted version of Hill et al's conjecture and finally gives the desired outcome. Hence, decomposing <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> into a cluster sequence with respect to Θ = subtree-like can be used to speed up the exact calculation of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M3">View MathML</a>.

However, for practical problem instances, it may be unlikely to find many subtree-like clusters. For example, the two phylogenies shown in Figure 1 do not have any common subtree-like cluster. This is due to the restricted definition of such a cluster which requires that a vertex whose set of descendants is a common cluster of two rooted binary phylogenetic X-trees <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> has the same parent vertex than a common subtree of <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a>. To lessen this problem, an alternative approach--that has recently been published by Linz and Semple [6]--can be applied. This paper describes a more general divide-and-conquer approach that exactly computes the rSPR distance between <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> for when a cluster sequence <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M27">View MathML</a> with respect to Θ = minimal for <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M1">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M2">View MathML</a> is given. Loosely speaking, the authors calculate a so-called minimum-weight partition <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M62">View MathML</a> of X ∪ {ρ} ∪ {a1, a2, ..., at , ρ1, ρ2, ..., ρt} such that <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M62">View MathML</a> contains an agreement forest (not necessarily a maximum-agreement forest) for each tree pair (<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M39">View MathML</a>). To compute <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M62">View MathML</a>, it has been shown that applying a 'bottom-up' approach which locally works on subtrees of each tree pair (<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M39">View MathML</a>) guarantees that the number of elements in <a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M62">View MathML</a> that are subsets of {a1, a2, ..., at , ρ1, ρ2, ..., ρt} is maximized while |<a onClick="popup('http://www.biomedcentral.com/1471-2148/10/334/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/10/334/mathml/M62">View MathML</a>| is minimized.

Acknowledgements

I thank Maria Luisa Bonet, Mareike Fischer, and Charles Semple for useful discussions and comments on an earlier version of this paper. Financial support from MEC (TIN2007-68005-C04-03) is gratefully acknowledged.

Response

By Helgi B Schiöth

E-Mail: helgis@bmc.uu.se

Address: Department of Neuroscience, Functional Pharmacology, Uppsala University, BMC, Box 593, 751 24, Uppsala, Sweden

"We have found that the manuscript by Linz is correct and to the point. We have therefore updated the SPRIT software and published the new version online.

The new version supports both the old incorrect conjecture as well as the new correct one to allow for comparisons to be made."

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