Open Access

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 T https://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 T https://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 ( T ) https://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 T https://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 T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gifto X′, denoted by T https://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 T https://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 T https://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 T https://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 T https://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 v https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq3_HTML.gif. Let T https://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 T https://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 v https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif has been obtained from T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif into T https://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 d rSPR ( T , T ) https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq5_HTML.gif.

Agreement Forests

Let T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 = { p , 1 , 2 , ... , k } https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq6_HTML.gif for T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif is a partition of X { ρ } https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq7_HTML.gif such that ρ ρ https://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 T | i T | i https://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 { T ( i ) : i { ρ , 1 , ... , k } } https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq10_HTML.gif and { T ( i ) : i { ρ , 1 , ... , k } } https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 { T | ρ , T | 1 , T | 2 , ... , T | k } https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq12_HTML.gif and { ρ , 1 , ... , k } https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gifand T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gifbe two rooted binary phylogenetic X-trees, and let { T ρ , T 1 , T 2 , ... , T k } https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq14_HTML.gifbe a maximum-agreement forest for T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gifand T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif. Then
d rSPR ( T , T ) = k . https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equa_HTML.gif

Clusters

Let T https://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 T https://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 v https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq3_HTML.gif in T https://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 C T ( v ) https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 v https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq3_HTML.gif in T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and a vertex v https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq16_HTML.gif in T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif such that C T ( v ) = C T ( v ) https://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 v https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq3_HTML.gif (resp. v https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq16_HTML.gif) in T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif (resp. T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif), and let w (resp. w′) be the child vertex of u (resp. u′) with w v https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq18_HTML.gif (resp. w v https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 C T ( u ) = C T ( u ) https://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 T | C T ( w ) T | C T ( w ) https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq21_HTML.gif. Roughly speaking, the condition T | C T ( w ) T | C T ( w ) https://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 T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif. We refer to T | C T ( w ) https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 C T ( u ) = C T ( u ) https://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 T | ( 7 , 8 , , 12 ) T | ( 7 , 8 , , 12 ) https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif.
https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif. Note that T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 ( T ) = X { ρ } https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif with | ( T ) | | A i | > 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq25_HTML.gif. Let T i https://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 T | A i https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq27_HTML.gif (viewing the root of T i https://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 T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif by replacing T ( A i ) https://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 T i https://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 T | A i https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq30_HTML.gif (viewing the root of T i https://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 T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif by replacing T ( A i ) https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif contain a Θ cluster Ai+1 with | ( T ) | | A i + 1 | > 1 https://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
( T 1 , T 1 ) , ... , ( T t , T t ) , ( T ρ , T ρ ) https://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 T ρ https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and T ρ https://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 T ( A t ) https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq35_HTML.gif and T ( A t ) https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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.
https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif 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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gifand T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gifbe two rooted binary phylogenetic X-trees. Let ( T 1 , T 1 ) , ... , ( T t , T t ) , ( T ρ , T ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq37_HTML.gifbe a cluster sequence for T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gifand T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gifwith respect to Θ = solvable. Then
d rSPR ( T , T ) = i = 1 t d rSPR ( T i , T i ) + d rSPR ( T ρ , T ρ ) . https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equ1_HTML.gif
(1)
Next, we detail a counterexample to the above conjecture which is based on the two rooted binary phylogenetic X-trees T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq38_HTML.gif for T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 d rSPR ( T , T ) = 4 https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 ( T 1 , T 1 https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 A1 = {1, 2, ..., 6} of T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif; thus T 1 = T | ( A 1 { ρ 1 } ) https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq41_HTML.gif and T 1 = T | ( A 1 { ρ 1 } ) https://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 ( T 2 , T 2 https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif by replacing T ( A 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq44_HTML.gif and T ( A 1 ) https://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 a1 whose leaf set is the solvable cluster A2 = {7, 8, ..., 12}. Lastly, the third tree pair ( T ρ , T ρ https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif by replacing T ( A 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq44_HTML.gif and T ( A 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq45_HTML.gif with a leaf a1 and replacing T ( A 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq47_HTML.gif and T ( A 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq48_HTML.gif with a leaf a2. For each tree pair ( T i , T i https://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 i https://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 i https://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 T i https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq26_HTML.gif and T i https://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
https://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 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq38_HTML.gif for T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 i https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq50_HTML.gif for each tree pair T i https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq26_HTML.gif and T i https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq29_HTML.gif shown in Figure 2.

d rSPR ( T 1 , T 1 ) + d rSPR ( T 2 , T 2 ) + d rSPR ( T ρ , T ρ ) = 2 + 2 + 1 = 5 https://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 d rSPR ( T , T ) https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gifand T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gifbe two rooted binary phylogenetic X-trees. Let ( T 1 , T 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq40_HTML.gif), ( T ρ , T ρ https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gifand T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gifwith respect to Θ = subtree-like. Then
d rSPR ( T , T ) = d rSPR ( T 1 , T 1 ) + d rSPR ( T ρ , T ρ ) . https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equd_HTML.gif

Proof. Let A1 be the subtree-like cluster ( T 1 ) { ρ 1 } https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq51_HTML.gif of T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 S https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq52_HTML.gif, that is associated with A1 in T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif. Clearly, S https://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 T ρ https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and T ρ https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq34_HTML.gif. Furthermore, as T ρ https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif has been obtained from T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif by replacing T ( A 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq44_HTML.gif with a single vertex a1 and as T ρ https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq34_HTML.gif has been obtained from T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif by replacing T ( A 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq45_HTML.gif with a single vertex a1, it is easily checked that T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif|( ( S ) { a 1 } https://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 T ρ https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and T ρ https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq34_HTML.gif.

We now show that
d rSPR ( T , T ) d rSPR ( T 1 , T 1 ) + d rSPR ( T ρ , T ρ ) . https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equ2_HTML.gif
(2)
Let 1 https://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 T 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq55_HTML.gif and T 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq56_HTML.gif, and let ρ https://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 T ρ https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and T ρ https://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 ( S ) { a 1 } https://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 a 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq58_HTML.gif, in ρ https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq57_HTML.gif. Furthermore, let ρ 1 https://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 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq54_HTML.gif with ρ1 ρ 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq59_HTML.gif. As 1 https://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 T 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq55_HTML.gif and T 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq56_HTML.gif and as ρ https://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 T ρ https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and T ρ https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq34_HTML.gif, it follows that
= ( 1 ρ { ρ 1 , a 1 } ) { ( ρ 1 { ρ 1 } ) ( a 1 { a 1 } ) } https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif. As a 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq58_HTML.gif - {a1} always contains an element, note that ( ρ 1 { ρ 1 } ) ( a 1 { a 1 } ) https://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 | | = | 1 | + | ρ | 1 https://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
d rSPR ( T 1 , T 1 ) + d rSPR ( T ρ , T ρ ) = | 1 | 1 + | ρ | 1 = | | 1 d rSPR ( T , T ) . https://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
d rSPR ( T , T ) d rSPR ( T 1 , T 1 ) + d rSPR ( T ρ , T ρ ) . https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_Equ3_HTML.gif
(3)
Let https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq38_HTML.gif, say m https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq62_HTML.gif, such that m A 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq63_HTML.gif and m ( X A 1 ) { ρ } https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq64_HTML.gif. Note that m https://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, https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif. Let m = ( m A 1 ) { ρ 1 } https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq65_HTML.gif, and let m ' ' = ( m ( ( X A 1 ) { ρ } ) ) { a 1 } https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq66_HTML.gif. Since https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif,
1 = { : A 1 } { m } https://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 T 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq55_HTML.gif and T 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq56_HTML.gif and
ρ = { : ( ( X A 1 ) { ρ } ) } { m ' ' } https://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 T ρ https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and T ρ https://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 m https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq62_HTML.gif exists. Hence, every element https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq67_HTML.gif in https://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 A1 or a subset of ( X A 1 ) { ρ } https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq68_HTML.gif. Furthermore, as A1 is a subtree-like cluster of T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 S https://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 ( S ) https://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 S https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq70_HTML.gif, in https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq38_HTML.gif. Now, as https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif, it follows that
1 = { : A 1 } { { ρ 1 } } https://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 T 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq55_HTML.gif and T 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq56_HTML.gif and
ρ = ( { : ( ( X A 1 ) { ρ } ) } { S } ) { S { a 1 } } https://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 T ρ https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq33_HTML.gif and T ρ https://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 m https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq62_HTML.gif exists, we have | | = | 1 | + | ρ | 1 https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq61_HTML.gif, and therefore,
d rSPR ( T , T ) = | | 1 = | 1 | + | ρ | 2 d rSPR ( T 1 , T 1 ) + d rSPR ( T ρ , T ρ ) . https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gifand T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gifbe two rooted binary phylogenetic X-trees. Let ( T 1 , T 1 ) , ... , ( T t , T t ) , ( T ρ , T ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq37_HTML.gifbe a cluster sequence for T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gifand T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gifwith respect to Θ = subtree-like. Then
d rSPR ( T , T ) = i = 1 t d rSPR ( T i , T i ) + d rSPR ( T ρ , T ρ ) . https://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 ( T 1 , T 1 ) , ... , ( T t , T t ) , ( T ρ , T ρ ) https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 ( T i , T i https://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 G https://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 {a1, a2, ..., a t , ρ1, ρ2, ..., ρ t }. In the case of our counterexample,
G = { { 1 , 2 , 3 } , { 4 , 5 , 6 } , { 7 , 8 , 9 } , { 10 , 11 , 12 } , { 13 , 14 , 15 , ρ } , { a 1 , a 2 , ρ 1 , ρ 2 } } https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq4_HTML.gif while G https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif - {{a1, a2, ρ1, ρ2}} is precisely a maximum-agreement forest for T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 d rSPR ( T , T ) https://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 G https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif that are subsets of {a1, a2, ..., a t , ρ1, ρ2, ..., 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 G https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif is a subset of {a1, a2, ..., a t , ρ1, ρ2, ..., 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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 d rSPR ( T , T ) https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 ( T 1 , T 1 ) , ... , ( T t , T t ) , ( T ρ , T ρ ) https://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 T https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq1_HTML.gif and T https://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 G https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif of X {ρ} {a1, a2, ..., a t , ρ1, ρ2, ..., ρ t } such that G https://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 ( T i , T i https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq49_HTML.gif). To compute G https://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 ( T i , T i https://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 G https://static-content.springer.com/image/art%3A10.1186%2F1471-2148-10-334/MediaObjects/12862_2010_Article_1549_IEq71_HTML.gif that are subsets of {a1, a2, ..., a t , ρ1, ρ2, ..., ρ t } is maximized while | G https://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: 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: 42-10.1186/1471-2148-10-42.PubMed CentralView ArticlePubMed
  2. Hein J, Jing T, Wang L, Zhang K: On the complexity of comparing evolutionary trees. Discrete Appl Math. 1996, 71: 153-169. 10.1016/S0166-218X(96)00062-5.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. 10.1007/s00285-005-0315-9.View ArticlePubMed
  4. Semple C, Steel M: Phylogenetics. 2003, Oxford University Press
  5. Bordewich M, Semple C: On the computational complexity of the rooted subtree prune and regraft distance. Ann Comb. 2004, 8: 409-423. 10.1007/s00026-004-0229-z.View Article
  6. Linz S, Semple C: A cluster reduction for computing the subtree distance between phylogenies. Ann Comb

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