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

BMC Evolutionary Biology201010:334

DOI: 10.1186/1471-2148-10-334

Received: 21 June 2010

Accepted: 29 October 2010

Published: 29 October 2010

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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and is frequently denoted by http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq2_HTML.gif . Furthermore, let X′ be a subset of X. The minimal rooted subtree of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif that connects all the leaves in X′ is denoted by http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif (X′) while the restriction of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif to X′, denoted by http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif |X′, is the rooted binary phylogenetic X′-tree obtained from http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif (X′) by contracting all degree-two vertices apart from the root.

Rooted Subtree Prune and Regraft

Let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif be a rooted binary phylogenetic X-trees. For the purposes of the upcoming definition, we view the root of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif as a vertex ρ adjoined to the original root by a pendant edge. Now, let e = {u, v} be any edge of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif that is not incident with ρ such that u is the vertex on the path from ρ to http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq3_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif be the rooted binary phylogenetic X-tree obtained from http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq3_HTML.gif with f , and contract u. Then http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif has been obtained from http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif by a rooted subtree prune and regraft (rSPR) operation. The rSPR distance between two rooted binary phylogenetic X-trees http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif is the minimum number of rSPR operations that transform http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif into http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif . We denote this distance by http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq5_HTML.gif .

Agreement Forests

Let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif be two rooted binary phylogenetic X-trees. Again, to make the following work, regard the roots of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif as a vertex ρ adjoined to the original root by a pendant edge. An agreement forest http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq6_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif is a partition of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq7_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq8_HTML.gif and the following properties are satisfied:
  1. (i)

    for all i ∈ {ρ, 1, ..., k}, we have http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq9_HTML.gif , and

     
  2. (ii)

    the trees in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq10_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq11_HTML.gif are vertex-disjoint subtrees of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif , respectively.

     

Throughout the remainder of this note, we will interchangeably refer to http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq12_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq13_HTML.gif as an agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif . A maximum-agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif is an agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif with the smallest number of elements over all agreement forests for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif . Note that a maximum-agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif be two rooted binary phylogenetic X-trees, and let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq14_HTML.gif be a maximum-agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equa_HTML.gif

Clusters

Let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif if there is a vertex http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq3_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif whose set of descendants is precisely A. We denote this cluster by http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq15_HTML.gif .

We next consider several different types of clusters that will play an important role in the remainder of this paper. Let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif be two rooted binary phylogenetic X-trees, and let A be a cluster that is common to http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif ; that is there exists a vertex http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq3_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and a vertex http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq16_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq17_HTML.gif . Furthermore, let u (resp. u′) be the parent vertex of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq3_HTML.gif (resp. http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq16_HTML.gif ) in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif (resp. http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq18_HTML.gif ), and let w (resp. w′) be the child vertex of u (resp. u′) with http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq18_HTML.gif (resp. http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq19_HTML.gif ). If no proper subset of A is a common cluster of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif , we refer to A as a minimal cluster. Moreover, A is a solvable cluster if A is minimal and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq20_HTML.gif . Lastly, we say that A is a subtree-like cluster if A is a solvable cluster and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq21_HTML.gif . Roughly speaking, the condition http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq21_HTML.gif is satisfied if the subtree with root w in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif is identical to the subtree with root w′ in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif . We refer to http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq22_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif that are shown in Figure 1 since http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq20_HTML.gif = {1, 2, ..., 12}. However, as http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq23_HTML.gif , it follows that A is not a subtree-like cluster of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif .
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Fig1_HTML.jpg
Figure 1

Two rooted binary phylogenetic X -trees and . Note that http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif into smaller subtrees. As previously, view the roots of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif as a vertex ρ adjoined to the original root by a pendant edge, and regard ρ as part of the label set; that is http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq24_HTML.gif . Setting i to be 1, let A i be a common Θ cluster of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq25_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq26_HTML.gif denote the rooted binary phylogenetic tree http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq27_HTML.gif (viewing the root of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq26_HTML.gif as a vertex ρ i adjoined to the original root by a pendant edge) and reset http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif to be the tree obtained from http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif by replacing http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq28_HTML.gif with a new vertex a i . Analogously, let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq29_HTML.gif denote the rooted binary phylogenetic tree http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq30_HTML.gif (viewing the root of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq29_HTML.gif as a vertex ρ i adjoined to the original root by a pendant edge) and reset http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif to be the tree obtained from http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif by replacing http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq31_HTML.gif with a new vertex a i . If http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif contain a Θ cluster A i+1 with http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq32_HTML.gif , stop or increment i by 1 and repeat this process; otherwise, stop. Eventually, we obtain a sequence
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equb_HTML.gif
of pairs of rooted binary phylogenetic trees, where http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq34_HTML.gif denote the two trees after the replacement of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq36_HTML.gif with a vertex a t . We call this sequence a cluster sequence of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif 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.
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Fig2_HTML.jpg
Figure 2

A cluster sequence with respect to Θ = solvable for the two rooted binary phylogenetic X -trees and 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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif be two rooted binary phylogenetic X-trees. Let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq37_HTML.gif be a cluster sequence for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif with respect to Θ = solvable. Then
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equ1_HTML.gif
Next, we detail a counterexample to the above conjecture which is based on the two rooted binary phylogenetic X-trees http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif that are shown in Figure 1. A maximum-agreement forest http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq38_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif contains 5 elements and is shown in the top of Figure 3. By Theorem 1, this implies that http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq39_HTML.gif . Now, consider the cluster sequence with respect to Θ = solvable for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif that contains three tree pairs and is depicted in Figure 2. The first tree pair ( http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq40_HTML.gif ) consists of the restricted subtrees of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif whose leaf set is the solvable cluster A 1 = {1, 2, ..., 6} of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif ; thus http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq42_HTML.gif . Similarly, the second tree pair ( http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq43_HTML.gif ) consists of the restricted subtrees of the two trees that have been obtained from http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif by replacing http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq44_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq45_HTML.gif , respectively, with a single leaf a 1 whose leaf set is the solvable cluster A 2 = {7, 8, ..., 12}. Lastly, the third tree pair ( http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq46_HTML.gif ) can be regarded as being obtained from http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif by replacing http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq44_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq45_HTML.gif with a leaf a 1 and replacing http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq47_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq48_HTML.gif with a leaf a 2. For each tree pair ( http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq49_HTML.gif ) of the cluster sequence shown in Figure 2, a maximum-agreement forest http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq50_HTML.gif with i ∈ {1, 2, ρ} is depicted in the bottom part of Figure 3. Note that each forest http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq50_HTML.gif is the unique maximum-agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq26_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq29_HTML.gif Now, by Equation 1, we have
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Fig3_HTML.jpg
Figure 3

Maximum-agreement forests. Top: A maximum-agreement forest http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq38_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif depicted in Figure 1. Bottom: A maximum-agreement forest http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq50_HTML.gif for each tree pair http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq26_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq29_HTML.gif shown in Figure 2.

http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equc_HTML.gif

which is strictly greater than http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq5_HTML.gif ; 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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif be two rooted binary phylogenetic X-trees. Let ( http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq40_HTML.gif ), ( http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq46_HTML.gif ) be a cluster sequence for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif with respect to Θ = subtree-like. Then
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equd_HTML.gif

Proof. Let A 1 be the subtree-like cluster http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq51_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq52_HTML.gif , that is associated with A 1 in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif . Clearly, http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq52_HTML.gif is also a common subtree of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq34_HTML.gif . Furthermore, as http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif has been obtained from http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif by replacing http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq44_HTML.gif with a single vertex a 1 and as http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq34_HTML.gif has been obtained from http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif by replacing http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq45_HTML.gif with a single vertex a 1, it is easily checked that http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif |( http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq53_HTML.gif ) is a common subtree of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq34_HTML.gif .

We now show that
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equ2_HTML.gif
Let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq54_HTML.gif be a maximum-agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq56_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq57_HTML.gif be a maximum-agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq34_HTML.gif . By the observation prior to this paragraph, it follows from Proposition 3.2 of [5] that http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq53_HTML.gif is a subset of an element, say http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq58_HTML.gif , in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq57_HTML.gif . Furthermore, let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq59_HTML.gif be the label set of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq54_HTML.gif with ρ 1 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq59_HTML.gif . As http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq54_HTML.gif is an agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq56_HTML.gif and as http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq57_HTML.gif is such a forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq34_HTML.gif , it follows that
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Eque_HTML.gif
is an agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif . As http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq58_HTML.gif - {a 1} always contains an element, note that http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq60_HTML.gif is never the empty set. Thus http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq61_HTML.gif and, by Theorem 1, we have
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equf_HTML.gif

This establishes Equation 2.

We now turn to the second part of this proof and show that
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equ3_HTML.gif
Let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq38_HTML.gif be a maximum-agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif . The remainder of this part splits into two cases. First, assume that there exists an element in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq38_HTML.gif , say http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq62_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq63_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq64_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq62_HTML.gif is the only label set with the described properties, as otherwise, http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq38_HTML.gif is not an agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq65_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq66_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq38_HTML.gif is an agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equg_HTML.gif
is such a forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq56_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equh_HTML.gif
is an agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq34_HTML.gif . Second, assume that no such element http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq62_HTML.gif exists. Hence, every element http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq67_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq38_HTML.gif is either a subset of A 1 or a subset of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq68_HTML.gif . Furthermore, as A 1 is a subtree-like cluster of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif whose associated common subtree is http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq52_HTML.gif , it again follows from Proposition 3.2 of [5], that http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq69_HTML.gif is a subset of an element, say http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq70_HTML.gif , in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq38_HTML.gif . Now, as http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq38_HTML.gif is an agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif , it follows that
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equi_HTML.gif
is an agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq56_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equj_HTML.gif
is such a forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq34_HTML.gif . Regardless of whether or not http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq62_HTML.gif exists, we have http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq61_HTML.gif , and therefore,
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equk_HTML.gif

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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif be two rooted binary phylogenetic X-trees. Let http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq37_HTML.gif be a cluster sequence for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif with respect to Θ = subtree-like. Then
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equl_HTML.gif

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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq37_HTML.gif be a cluster sequence with respect to Θ = solvable for two rooted binary phylogenetic trees http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif . Calculating a maximum-agreement forest for each tree pair ( http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq49_HTML.gif ), taking their union, and, for each i ∈; {1, 2, ..., t}, joining the element containing a i with the element containing ρ i can potentially result in a set, say http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif , which contains an element that is a subset of {a 1, a 2, ..., a t , ρ 1, ρ 2, ..., ρ t }. In the case of our counterexample,
http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equm_HTML.gif

contains one such element. Trivially, this element is not part of any agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif while http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif - {{a 1, a 2, ρ 1, ρ 2}} is precisely a maximum-agreement forest for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif . Consequently, a divide-and-conquer approach that exactly calculates http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq5_HTML.gif needs to take into account the number of elements in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif that are subsets of {a 1, a 2, ..., a t , ρ 1, ρ 2, ..., ρ t }; 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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif is a subset of {a 1, a 2, ..., a t , ρ 1, ρ 2, ..., ρ t }. 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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif into a cluster sequence with respect to Θ = subtree-like can be used to speed up the exact calculation of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq5_HTML.gif .

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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif has the same parent vertex than a common subtree of http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif for when a cluster sequence http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq37_HTML.gif with respect to Θ = minimal for http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif is given. Loosely speaking, the authors calculate a so-called minimum-weight partition http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif of X ∪ {ρ} ∪ {a 1, a 2, ..., a t , ρ 1, ρ 2, ..., ρ t } such that http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif contains an agreement forest (not necessarily a maximum-agreement forest) for each tree pair ( http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq49_HTML.gif ). To compute http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif , it has been shown that applying a 'bottom-up' approach which locally works on subtrees of each tree pair ( http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq49_HTML.gif ) guarantees that the number of elements in http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif that are subsets of {a 1, a 2, ..., a t , ρ 1, ρ 2, ..., ρ t } is maximized while | http://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif | is minimized.

Declarations

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: http://​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."

Authors’ Affiliations

(1)
Department of Computer Science, Technical University of Catalonia

References

  1. Hill T, Nordström KJV, Thollesson M, Säfström TM, Vernersson AKE, Fredriksson R, Schiöth HB: SPRIT: Identifying horizontal gene transfer in rooted phylogenetic trees. BMC Evol Biol 2010, 10:42.PubMedView Article
  2. Hein J, Jing T, Wang L, Zhang K: On the complexity of comparing evolutionary trees. Discrete Appl Math 1996, 71:153–169.View Article
  3. Baroni M, Grünewald S, Moulton V, Semple C: Bounding the number of hybridization events for a consistent evolutionary history. J Math Biol 2005, 51:171–182.PubMedView Article
  4. Semple C, Steel M: Phylogenetics. Oxford University Press; 2003.
  5. Bordewich M, Semple C: On the computational complexity of the rooted subtree prune and regraft distance. Ann Comb 2004, 8:409–423.View Article
  6. Linz S, Semple C: A cluster reduction for computing the subtree distance between phylogenies. Ann Comb, in press.

Copyright

© Linz. 2010

This article is published under license to 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.