Refinement-stable Consensus Methods
aa r X i v : . [ q - b i o . P E ] F e b Refinement-stable Consensus Methods
Mareike Fischer , Michael Hendriksen Institute of Mathematics and Computer Science, University ofGreifswald, Germany Institut f¨ur Molekulare Evolution, Heinrich-Heine Universit¨at,D¨usseldorf, Germany
Abstract
In a recent study, Bryant, Francis and Steel investigated the concept of“future-proofing” consensus methods in phylogenetics. That is, they inves-tigated if such methods can be robust against the introduction of additionaldata like extra trees or new species. In the present manuscript, we analyzeconsensus methods under a different aspect of introducing new data, namelyconcerning the discovery of new clades. In evolutionary biology, often formerlyunresolved clades get resolved by refined reconstruction methods or new ge-netic data analyses. In our manuscript we investigate which properties ofconsensus methods can guarantee that such new insights do not disagree withpreviously found consensus trees but merely refine them. We call consensusmethods with this property refinement-stable . Along these lines, we also studytwo famous super tree methods, namely Matrix Representation with Parsi-mony (MRP) and Matrix Representation with Compatibility (MRC), whichhave also been suggested as consensus methods in the literature. While we(just like Bryant, Francis and Steel in their recent study) unfortunately haveto conclude some negative answers concerning general consensus methods,we also state some relevant and positive results concerning the majority rule(MR) and strict consensus methods, which are amongst the most frequentlyused consensus methods. Moreover, we show that there exist infinitely manyconsensus methods which are refinement-stable and have some other desirableproperties.
In phylogenetics, consensus methods play a fundamental role concerning tree re-construction: For instance when different genes of the same set of species lead todifferent gene trees or when different tree reconstruction methods come to differentresults, it may be hard to decide which of the given trees is the “true” tree in thesense of coinciding with the underlying (unknown) species tree. This is where con-sensus methods come into play — they use certain rules to summarize a set of trees1o form a consensus tree . There are various such methods used in biology, and theirresults can be quite different. However, even if you stick to one consensus method,its outcome might change when new input data is discovered. This is why it isimportant to determine how “future-proof” consensus methods are; i.e. how robustthey are against the introduction of new data.In a recent study [6], Bryant, Francis and Steel investigated the properties ofconsensus methods in an axiomatic manner, proposing three simple conditions thata consensus method should obey, referring to any such method as regular . Thepurpose of the article was precisely to investigate the concept of “future-proofing”with consensus methods. In particular, they investigated associative stability –robustness against the introduction of additional trees, and extension stability –robustness against the introduction of additional species. Unfortunately, it hasrecently been determined that such future-proofing is impossible, as there exist noregular, extension stable consensus methods [6], and while regularity and associativestability is possible, a consensus method cannot be regular, associatively stable andPareto on rooted triples [8], i.e. the combination of certain desirable properties isnot possible.In the present paper we investigate a related question – can a consensus methodbe robust against refinement of the input trees? This question is of the utmost impor-tance, as often formerly unresolved clades in known phylogenetic trees get resolvedby new genetic analyses or refined tree reconstruction methods. This implies thata given set of input trees might be changed in the sense that new clades are addedrather than new species or entirely new trees as in [6], and the main purpose of thepresent manuscript is to analyze the impact of this scenario on consensus methods.Additionally, we also study two so-called supertree methods, namely Matrix Rep-resentation with Parsimony (MRP) and Matrix Representation with Compatibility.These methods are sometimes considered as consensus methods in the literature,even though there is some ongoing debate about that [2, 3], because these methodsdo not necessarily lead to a unique tree and thus require another consensus methodto summarize all trees they find. However, we will show in the present manuscriptthat even in the ideal case in which these methods do lead to a unique tree, thesemethods are not future-proof – neither in the sense of adding new species (thuscomplementing the study of [6]) nor in the sense of resolving new clades.While the above mentioned findings might be considered “bad news”, we alsoshow that majority rule consensus methods (including the so-called strict consensus)are indeed refinement-stable , i.e. they are robust against the new resolution offormerly unresolved input trees. Moreover, we are even able to show that there existinfinitely many consensus methods which are regular and refinement-stable, and evennon-contradictory (which is yet another desirable property) – even if the class of suchmethods does not contain some of the established methods like loose consensus orAdams consensus. This paves the way for new directions in future research, namelythe the search for new consensus methods which have these desirable properties andare biologically plausible. Such methods could have huge potential in replacing someof the existing consensus methods. 2
Preliminaries
Before we can present our results, we need to formally introduce the most importantconcepts discussed in this manuscript.
In the following, let X be a finite set, typically a set of taxa or species , but forsimplicity, we may also assume without loss of generality that, whenever | X | = n ,we have X = { , . . . , n } . Recall that a phylogenetic X -tree T is a connected acyclicgraph whose leaves are bijectively labelled by the elements of X . If there is onedistinguished node ρ referred to as the root of the tree, then T is called rooted ;otherwise T is called unrooted . Let RP ( X ) and U P ( X ) denote the set of rooted andunrooted phylogenetic trees on X , respectively. Then, a profile of trees is an orderedtuple ( T , ..., T k ) of trees such that T , ..., T k ∈ RP ( X ) or T , ..., T k ∈ U P ( X ) (thatis, the trees in a profile must all be rooted or all be unrooted, and they must allrefer to the same taxon set X ).We now first turn our attention to unrooted trees. Recall that a bipartition σ of X into two non-empty and disjoint subsets A and B is often called X -split (or split for short if there is no ambiguity), and is denoted by σ = A | B . Also recallthat there is a natural relationship between X -splits and the edges of an unrootedphylogenetic X -tree T , because the removal of an edge e induces such a bipartition of X . In the following, the set of all such induced X -splits of T will be denoted by Σ( T ).Moreover, note that the size of an X -split σ = A | B is defined as | σ | = min {| A | , | B |} [11]. An X -split of size 1 is called trivial . Note that two X -splits σ = A | B and σ = e A | e B are called compatible if at least one of the intersections A ∩ e A , A ∩ e B , B ∩ e A or B ∩ e B is empty (note that if more than one of them is empty, then σ = σ ).A fundamental and classic insight concerning unrooted trees is provided by thefollowing classic theorem by Buneman [7], which is also known as Splits-EquivalenceTheorem (see also [14, p. 44]). Theorem 2.1 (Buneman) . Let Σ be a collection of X -splits. Then, there is anunrooted phylogenetic X -tree T such that Σ = Σ( T ) if and only if Σ contains thetrivial splits and all splits in Σ are pairwise compatible. Moreover, if such a treeexists, it is unique (up to isomorphism). The Buneman theorem implies that an unrooted phylogenetic X -tree T is uniquelydetermined by its non-trivial X -splits Σ ∗ ( T ) (note that given Σ ∗ ( T ), the uniquetree T can be found by the so-called Tree Popping algorithm [13, 14]). Recall thatthese splits can be coded as binary characters B ∗ ( T ). A character is a function f : X −→ C from X to some alphabet C , and it is called binary whenever | C | = 2.For simplicity, in such cases we assume C = { , } . Note that the elements of analphabet are also often referred to as states or character states . Moreover, note thatwe often use the shorthand f (1) f (2) . . . f ( n ) for a character on X = { , . . . , n } . Forinstance, rather than explicitly writing f (1) = 1, f (2) = 1, f (3) = 0 and f (4) = 0,3 igure 1: All phylogenetic X -trees for X = { , , , } . we will write f = 1100. Note that the characters 1100 and 0011 refer to the same X − split and are thus considered equivalent. Therefore, we will assume withoutloss of generality that when we translate a tree to its non-trivial X -splits and these X -splits in turn to binary characters, taxon 1 is always in state 1.As an example, consider the four 4-taxon trees on X = { , , , } depictedin Figure 1. Tree T , which has no non-trivial splits, has Σ ∗ ( T ) = ∅ and thusalso B ∗ ( T ) = ∅ . On the other hand, T has the Σ ∗ ( T ) = { | } , and thus B ∗ ( T ) = { } .For rooted trees it is known that they are not fully determined by their corre-sponding sets of splits, because these do not contain any information on the positionof the root. Thus, instead of considering splits, in the rooted setting we consider clusters . Recall that a clade of a rooted phylogenetic tree T is a pendant subtree T ′ of T , and a cluster is the set of leaves Y ⊆ X of T ′ . Clusters of sizes 1, i.e.referring to single leaves, and n , i.e. referring to all leaves X , are called trivial .We denote the set of all clusters of a rooted phylogenetic tree T by C ( T ), and theset of its non-trivial clusters by C ∗ ( T ). Note that every cluster Y ⊂ X (i.e. everystrict subcluster of X ) induces an X -split, namely σ = Y | ( X \ Y ). In the following,we call two clusters c , c ∈ C ( T ) of a rooted phylogenetic X -tree compatible if theinduced splits σ and σ of c and c , respectively, are compatible and if additionallywe have c ⊆ c or c ⊆ c or c ∩ c = ∅ . Note that the latter condition is requiredbecause while, for instance, the splits 123 |
456 and 12 | { , , } ∩ { , , } = ∅ ), the clusters c = { , , } and c = { , , , } , whichinduce these splits, do not fit together on a rooted tree.Now, with these definitions, can be easily seen that, just as an unrooted phylo-genetic tree T is fully determined by Σ ∗ ( T ), a rooted phylogenetic tree T is fullydetermined by its set of clusters, which we denote by C ∗ ( T ). In fact, this is a direct4onsequence of the Buneman theorem: Corollary 2.1 (Buneman-type theorem for rooted trees) . Let C be a collection ofclusters on X . Then, there is a phylogenetic X -tree T such that C = C ( T ) if andonly if the clusters in C are pairwise compatible. Moreover, if such a tree exists, itis unique (up to isomorphism).Proof. By the definition of compatible clusters, it is clear that – as their inducedsplits must be compatible, too – by Theorem 2.1, they correspond to a unique un-rooted tree T . So we only need to show that the root position is uniquely determinedby C , too. As we defined compatible clusters to either be nested or disjoint, we canfind all maximal clusters in C , i.e. all clusters which are not contained in any othercluster. These clusters must belong to vertices adjacent to the root, because if theroot was placed in any other vertex of T , at least one of these clusters would bebroken up. This completes the proof.Note that the unique rooted phylogenetic tree belonging to a set C of pairwisecompatible clusters can be easily found with the so-called BUILD algorithm [1].Before we can turn our attention to consensus methods, we need to introducetwo more pieces of notation. The first concept is the so-called Newick format. TheNewick format (cf. [9]) uses nested brackets in such a way that two closely relatedspecies are grouped closely together. For instance, if T = ((1 , , (3 , , T is rooted, i.e. for T = ((1 , , (3 , ,
5) this would imply that the root leads to thethree clusters { , } , { , } and { } . However, if T is unrooted, the root informationinherent to the Newick format is simply disregarded.The last concept we need to recall before we can turn our attention to consensusmethods is the relation (cid:22) . For two rooted (or unrooted) phylogenetic trees T and T , we say T refines T and denote this by T (cid:22) T , whenever C ( T ) ⊆ C ( T ) (orΣ( T ) ⊆ Σ( T ), respectively). It can be easily seen that (cid:22) is a partial order. As anexample, in Figure 1, we have T (cid:22) T , T (cid:22) T and T (cid:22) T , but there is no suchrelation between T , T and T . We are now finally in the position to introduce the most important concept under-lying this manuscript.
Definition 2.2. A rooted consensus method (resp. unrooted consensus method ) isa function φ that, for every set X of taxa and every number k ≥
1, associates witheach profile of k trees from RP ( X ) (resp. U P ( X )) a unique corresponding tree in RP ( X ) (resp. U P ( X )).Following [6], we call a consensus method φ regular if it obeys the following threeproperties: 5. Unanimity:
If the trees in P are all the same tree T , then φ ( P ) = T .2. Anonymity:
Changing the order of the trees in P does not change φ ( P ).3. Neutrality:
Changing the labels on the leaves of the trees in P simply relabelsthe leaves of φ ( P ) in the same way.Unanimity reflects the completely reasonable assertion that if all of your datapoints towards a given tree, then the data is best represented by that particulartree. It additionally prevents useless consensus methods, such as φ ( P ) = T for all P , given some T . Anonymity means that the order does not affect the consensusmethod, preventing more useless consensus methods, such as returning the first treein the profile every time. Neutrality reflects the condition that the labels shouldnot affect the outcome – that is, if you swap two species, say cat and dog , in everytree in the profile, the only outcome should be that cat and dog are swapped in theconsensus tree.We now want the reader to recall three of the most frequently used consensusmethods. Definition 2.2.
Let P be a profile of trees ( T , . . . , T k ). Then, we define the fol-lowing rules for forming a consensus tree: • Strict consensus ( Γ ): The strict consensus tree Γ( P ) contains precisely allclusters (in the rooted setting), respectively all splits (in the unrooted setting)that are present in all trees T i ∈ P . • Loose (or semi-strict) consensus ( γ ): The loose consensus tree γ ( P )contains precisely all clusters (or splits, respectively) that are present in atleast one tree T i ∈ P and that are not incompatible with any split induced byany tree in P . • Majority rule consensus (
M R ): Let p >
50. Then, the majority-ruleconsensus tree
M R p ( P ) contains precisely all clusters (or splits, respectively)that are present in at least p % of the trees T i ∈ P . Note that whenever thereis no ambiguity concerning p or whenever a statement holds for all possiblechoices of p >
50, we may simply write
M R ( P ) rather than M R p ( P ).Before we can continue with a remark on how these methods are related, we needto state our first lemma. Lemma 2.3.
In all above cases, the respective consensus tree φ ( P ) exists and isunique.Proof. We start by considering the unrooted case. We need to show that for allthree definitions, there is precisely one tree φ ( P ) fulfilling the stated conditions. Asthe splits that shall be contained are in all cases uniquely determined, we know byTheorem 2.1 that if these splits are pairwise compatible, the resulting tree φ ( P ) willbe unique. So the only thing that remains to be shown is the compatibility of thesplits used for constructing φ ( P ). 6 Majority rule consensus:
By definition, all splits that shall be used for
M R ( P ) appear in at least one of the trees of P , but these splits might becontradicted by some splits of other trees of P . However, as we chose k > M R ( P ) appears in more than half of the trees of P . This implies that for any two splits σ , σ thatare used for M R ( P ), there is at least one tree in P that induces both of them.Thus, they must be compatible, which in turn implies pairwise compatibilityof all splits used for M R ( P ). • Loose (or semi-strict) consensus:
By definition, all splits that shall beused for γ ( P ) appear in at least one of the trees of P and are not contradictedby any of the trees – which means they are not incompatible with any split ofany T i . So again by Theorem 2.1, all these splits are pairwise compatible. • Strict consensus:
By definition, we have Γ =
M R , i.e. strict consensusis just a special case of majority-rule consensus. So there remains nothing toshow.So in all cases, a set of pairwise compatible splits is used to build the consensus.Thus, by Theorem 2.1, the resulting tree T P exists and is unique. This completesthe unrooted case.The rooted case follows analogously by using clusters instead of splits and Corol-lary 2.1 instead of Theorem 2.1. This completes the proof.Next, we want to introduce two more classical consensus methods, namely theAdams and the Aho consensus. Our definitions are based on [6], but for furtherdetails, we refer the reader also to [5].We start with Adams consensus . In order to build the Adams consensus treefor a profile P of phylogenetic X -trees, we start by considering the partition Π( X ),which equals the non-empty intersections of the maximal clusters of the trees in P .Note that Π( X ) will correspond to the maximal clusters in the consensus tree. Oncethis partition Π( X ) is determined, we take each element of Π( X ) and recursivelyrepeat this procedure for the respective subset of taxa until it has size one and thuscannot be refined anymore. This eventually produces a set of compatible clusterson X , which by Corollary 2.1 can be assembled into a unique tree.Similarly, for Aho consensus , we construct a partition Π( X ) that also getsrecursively refined, but in this case, Π( X ) equals the connected components of thegraph ( X, E P ), where there is an edge { x , x } ∈ E P precisely if there exists x ∈ X \ { x , x } such that the subtree (( x , x ) , x ) is contained in all trees of P .In both cases, the hierarchy induced by the recursive partitioning can be used toreconstruct a unique rooted tree (which we refer to as ϕ Ad ( P ) or ϕ Aho ( P ), respec-tively) using the famous BUILD algorithm [1]. Note, however, that both the Adamsand the Aho consensus are only defined in the rooted setting and do not work forunrooted trees. 7 emark 2.3. Recall that by [6, p. 613], “All standard phylogenetic consensus meth-ods (e.g., strict consensus, majority rule, loose consensus, and Adams consensus)”,are regular. It is easy to see that this, too, applies to Aho consensus.
Remark 2.4.
Let P be a profile of (rooted or unrooted) phylogenetic X -trees.Then, it can be easily seen by Definition 2.2 that Γ( P ) (cid:22) M R ( P ).One of the primary aims of this paper is to consider a new concept of future-proofing, namely that of refinement stability . Definition 2.5.
Let φ be a consensus method. If, for any pair of profiles P =( T , . . . , T k ) , P ′ = ( T ′ , . . . , T ′ k ) so that T i (cid:22) T ′ i for each i ∈ { , ..., k } , we have φ ( P ) (cid:22) φ ( P ′ ), then φ is referred to as refinement-stable .Another natural property that might be desirable for a consensus method is thatof being non-contradictory : Definition 2.6.
A consensus method is termed non-contradictory if each cluster(or split, respectively) of the output tree is compatible with at least one tree in theinput profile.We now turn our attention to two supertree methods, which also sometimesappear in the context of consensus trees.
In the literature you can find several well-known methods to construct so-calledsupertrees from (multi)sets of input trees. One difference to our setting is that theinput trees used to build a supertree need not coincide in their taxon sets. It is quitetypical, in fact, that their taxon sets do not coincide, but usually they overlap. Inthis case, the supertree will contain all taxa present in any of the input trees andcombine the information of the input trees as best as possible. For instance, thisapproach is used for reconstructing the Tree of Life, i.e. the tree of all living species[3, 4]. It is obvious, though, that in the supertree setting, too, input trees mightcome with conflicting information, in which case the supertree corresponds to somesort of consensus. This is why supertree methods have also often been regarded asconsensus methods by some authors [2, 5, 12], even if there is an ongoing debatewhether this is justified [2, 3].The reason why it is not straight forward to regard supertree methods as consen-sus methods is that (as opposed e.g. to strict consensus or majority-rule consensuswith k >
50) a supertree need not be unique. This problem can be overcome e.g. byusing an existing consensus method such as majority-rule or strict in order to sum-marize all supertrees into a single consensus tree [5]. However, another argumentagainst regarding supertree methods as consensus methods is that a supertree maycontain splits (or clusters) that are not present in any of the input trees. However,in the present manuscript, we will not join this debate, but we will consider twofamous supertree methods as consensus methods in the case in which the respective8upertree is unique and test them for refinement stability: Matrix Representationwith Parsimony , or
MRP for short, and
Matrix Representation with Compatibility ,or
MRC for short. Note that – as we only consider cases where the output of MRPand MRC is unique – applying a regular consensus method like majority-rule to theoutput of MRP or MRC as suggested for instance in [5] would not change anythingdue to unanimity.In order to understand MRP and MRC, we first need to introduce the concept of parsimony . In this regard, recall that an extension of a character f on a phylogenetic X -tree T with vertex set V ( T ) is a function g : V ( T ) −→ C , such that g f | X = f ,i.e. g agrees with f on the leaves of T but also assigns states to inner vertices of T . Now the changing number ch ( g f , T ) of an extension g f on a phylogenetic tree T is simply the number of edges e = { u, v } for which g f ( u ) = g f ( v ). An extension e g f such that e g f = min g f ch ( g f , T ) is called minimal , and the changing number ofsuch a minimal extension is called parsimony score of f on T , denoted ps ( f, T ).Thus, we have ps ( f, T ) = min g f ch ( g f , T ). The parsimony score of a character f ona phylogenetic tree T can for instance be calculated by the famous Fitch-Hartiganalgorithm . Moreover, the parsimony score of a (multi)set of characters, which isalso often referred to as alignment in evolutionary biology, is simply defined asthe sum of the parsimony scores of all characters. In particular, we can calculatethe parsimony score ps ( B ∗ ( T )) of B ∗ ( T ) of a phylogenetic X -tree T . Next, the maximum parsimony tree , or MP tree for short, for a (multi)set S of characters on X is a phylogenetic X -tree T for which we have ch ( S, T ) = ps ( S, T ), i.e. a treewhich has minimal changing number for S amongst all phylogenetic X -trees. Notethat this tree need not be unique.Given a profile P = ( T , . . . , T m ) of phylogenetic X -trees, we consider the union B := ∪ mi =1 B ∗ ( T i ). An MP tree of B is called Matrix Representation with Parsimonytree of P , or MRP tree of P for short (note that the word matrix in this contextstems from the fact that the 0-1-alignment B can also be regarded as a matrix).Again, the MRP tree need not be unique.The second supertree method which we want to consider is Matrix Representationwith Compatibility , or
MRC for short. This method is also based on alignment B .It analyzes all characters of B and finds a maximal compatible subset. Here, aset S of binary characters is called compatible if all characters in S are pairwisecompatible, and two binary characters are called compatible if their correspondingsplits are compatible. By Theorem 2.1, there is a unique tree corresponding toeach such set of compatible characters, and it can be easily found, for instancewith the so-called Tree Popping algorithm [13, 14]). In the following, we say that acompatible set of binary characters or splits induces T whenever T is the unique treecorresponding to this set according to Theorem 2.1. Now, concerning MRC: After a Note that the original Fitch algorithm is only suited for binary trees, but we explicitly allowfor non-binarity, which is why we refer to Hartigan’s generalization of this algorithm Note that biologists often regard B as an alignment. Then, order of characters in B is fixed,which is why it then makes sense to speak about a concatenation rather than a union. However, asthe order of characters does not play a role for the methods considered in the present manuscript,it is sufficient here to consider the union. B is found and the corresponding treeis constructed, this tree is called Matrix Representation with Compatibility tree or MRC tree for short. Note that there may be more than one MRC tree as theremight be more than one maximal compatible subset of characters in B .Note that both MRP and MRC can only reconstruct unrooted trees. This is dueto the fact that the root position neither has an impact on the parsimony score of acharacter on a given tree, nor on the compatibility of various splits (and thus theircorresponding binary characters). So while we generally consider both the rootedand the unrooted setting in this manuscript, this does not hold true whenever weconsider MRP or MRC, as these methods inherently work on unrooted trees only. In this section, we turn our attention to some established consensus methods andanalyze which of them are actually refinement-stable and which ones are not. Westart with a positive result. Note that whenever not stated otherwise, the resultshold both in the rooted as well as in the unrooted setting.
Theorem 3.1.
Strict consensus and majority-rule consensus are refinement-stable.Proof.
In the case of strict consensus, if only a few of the trees T i of P get refined,this does not immediately influence Γ( P ). Only if the refinement is such that a newsplit, say σ (or cluster, say c , respectively), is added to all input trees, then thissplit (or cluster) will also need to be added to Γ( P ). However, this is no problem,because as all splits (or clusters) of Γ( P ) are present in all input trees together with σ (or c ), this implies that they are all pairwise compatible both with one another aswell as with σ (or c , respectively). Thus, σ (or c ) can be added to Γ( P ) by Theorem2.1 (or by 2.1, respectively), which causes a refinement of this consensus tree, butno other modification.In the case of majority-rule consensus with k >
50, the situation is similar but abit more intricate. As long as fewer than 50% of the trees of P get refined such thatthey now contain a new split σ (or new cluster c ), this does not influence M R ( P ).If, however, more than k % >
50% of P contain a new split σ (or cluster c ) afterrefinement, this split (or cluster) has to be added to the consensus tree. However,this again must be possible. To see this, let τ (or c τ ) be a split (or cluster) alreadypresent in M R ( P ) before σ (or c ) is added. Then, by Definition 2.2, τ (or c τ ) iscontained in more than k % and thus in more than half of all trees of P , just as σ (or c ). So there must be at least one tree ˜ T which contains both σ and τ (or c and c τ ).Thus, by Theorem 2.1 (or Corollary 2.1), σ and τ (or c and c τ ) are compatible. As τ (or c ) was arbitrarily chosen amongst the splits (or clusters) of M R ( P ), all splitsof Σ( M R ( P )) (or all clusters of C ( M R ( P ))) are pairwise compatible with σ (or c ).10hus, again by Theorem 2.1 (or Corollary 2.1), σ (or c ) can get added to M R ( P )and will refine the consensus tree without any other modification. This completesthe proof.Next we consider the loose consensus method and state our first negative result. Proposition 3.1.
Loose consensus is not refinement-stable.Proof.
We prove this assertion by providing an explicit counterexample. Let X = { , , , , } and T = ((1 , , , ,
5) be either rooted or unrooted, and let S =(1 , , , ,
5) be the so-called (rooted or unrooted) star-tree, i.e. the unique tree on X with only one inner vertex. Let P = { T , S } . Then, for the loose consensus treewe have. γ ( P ) = T . However, if we refine S to become T = ((2 , , , ,
5) andconsider the profile P ′ = ( T , T ), we get γ ( P ′ ) = S (because the splits 12 |
345 and23 |
145 are incompatible). So if we refine input tree S , we get a coarser consensustree, not a refined one. This completes the proof.Now we turn our attention to the Adams and Aho consensus methods, whichlead to another negative result. Proposition 3.2.
Adams consensus and Aho consensus are not refinement-stable.Proof.
We prove the statement by presenting an explicit counterexample. Let T =(((1 , , , , T = (((1 , , (3 , ,
5) and T ′ = ((1 , , (3 , , P =( T , T ) and P ′ = ( T , T ′ ). Note that T ′ is a refinement of T , and hence anyconsensus method φ that is refinement-stable must have φ ( P ) (cid:22) φ ( P ′ ). However,the Adams consensus tree of P is tree T = ((1 , , , , P ′ is tree T ′ = ((1 , , , , not refinement-stable as ϕ ( P ) ϕ ( P ′ ).Note that in this example, Aho’s consensus coincides with Adams, so Aho’sconsensus method is also not refinement-stable, which completes the proof.So in summary, of the established consensus methods, strict and majority-ruleare refinement-stable, whereas loose consensus as well as Adams and Aho consensusare not. In Section 3.3 we will show that this also applies to MRP and MRC,unfortunately. But before we do so, we turn our attention to another property thatconsensus methods might have, namely that of being non-contradictory. Non-contradiction is another property that might be desirable for a biologicallymeaningful consensus method. It states that each cluster (or split) in the consensustree must be compatible with at least one input tree. The main aim of this shortsubsection is to state and prove that this is a property that indeed various establishedconsensus methods share, but not all of them.11 T T Figure 2: The Adams and Aho consensus methods are not non-contradictory as ϕ Ad ( T , T ) = ϕ Aho ( T , T ) = T . Proposition 3.3. γ , Γ and M R are all non-contradictory.Proof.
By Definition 2.2, all three methods lead to trees that only contain clusters(or splits) that are present in at least one tree. So each cluster (or split) in therespective consensus tree must be compatible with at least one tree in the inputprofile, namely with the one it is induced by. This completes the proof.Next, we consider Adams and Aho consensus.
Proposition 3.4.
Neither Adams nor Aho consensus are non-contradictory.Proof.
Consider trees T and T in Figure 2. For P = ( T , T ), Adams and Ahoconsensus coincide, and ϕ Ad ( P ) = ϕ Aho ( P ) = T , but T contains the clusters { , } and { , } , both of which are incompatible with both T and T . Hence neitherAdams nor Aho consensus are non-contradictory.We will now turn our attention to general consensus methods; i.e. we will nolonger just consider the established ones. In this section, we want to gather same general properties of refinement-stable con-sensus methods, whether these are established methods or not. On the one hand,12his will allow us to gain further insight into majority rule and strict consensus, twoof the most frequently used methods in phylogenetics, as both are refinement-stableby Theorem 3.1. On the other hand, however, general knowledge on refinement-stability also allows for conclusions on what to hope for concerning future consensusmethods – what properties can you wish for if you are aiming at a refinement-stablemethod?We have already seen that there are consensus methods that are not refinement-stable, such as the loose consensus. However, one can also construct other examplesof consensus methods which are not refinement-stable. For example, consider thefollowing: Define the rather trivial method in which the star tree is always returnedunless all trees in the profile are the same tree, in which case that tree is returned.Then this method is not refinement-stable. To see this, suppose the first profileconsists of two identical non-star trees that are not fully resolved, T . If we thenform the second profile by resolving exactly one of the trees more, the consensusbecomes the star tree.However, we start this section by showing the fundamental property that nec-essarily all refinement-stable and unanimous consensus methods refine the strictconsensus (note that by Theorem 3.1 this applies, for instance, to majority rule). Theorem 3.5.
Let φ be any refinement-stable, unanimous consensus method and Γ the strict consensus method. Let P be a profile of k trees. Then we have: Γ( P ) (cid:22) φ ( P ) . In particular, if a cluster c (or split σ in the unrooted case) is induced by alltrees in P , then c (or σ ) is also induced by φ ( P ) .Proof. Let the strict consensus of P = ( T , ..., T k ) be some tree T , i.e. T = Γ( P ).Then every tree in P contains each cluster (or each split, respectively) of T . It followsthat every tree in P is a refinement of T , and so given the profile Q := ( T, ..., T )consisting of k copies of T (where k ≥ T (cid:22) T i for each i in { , ..., k } ,and as φ ( Q ) = T by unanimity and φ ( Q ) (cid:22) φ ( P ), therefore T (cid:22) φ ( P ) and we havethe result.While the strict consensus thus is a coarse version of all possible consensus treesinduced by unanimous and refinement-stable methods, we now consider the oppositescenario, namely the loose consensus. Theorem 3.6.
Let φ be any refinement-stable, unanimous consensus method, andlet γ be the loose consensus method. Let P be a profile of k trees such that eachcluster (or each split, respectively) of every tree in P is compatible with all trees in P . Then φ ( P ) (cid:22) γ ( P ) .Proof. Let the loose consensus of P = ( T , ..., T k ) be the tree T := γ ( P ), notingthat by definition of loose consensus, every cluster (or split, respectively) of eachtree T i in P is contained in the clusters (or splits) of T as all these clusters (or splits)are by assumption compatible with all trees in P . It follows that, given the profile Q consisting of k copies of T , we have that T i (cid:22) T for each i in { , ..., k } , and as φ ( Q ) = T by unanimity and φ ( P ) (cid:22) φ ( Q ) = T = γ ( P ) by refinement-stability, wehave φ ( P ) (cid:22) γ ( P ), which completes the proof.13ote that Theorem 3.5 implies that the strict consensus tree is the coarsestrefinement of any refinement-stable and unanimous consensus method’s output tree.This, together with Theorem 3.6, might lead to the idea that maybe the opposite istrue for loose consensus: it might be the finest refinement of any refinement-stableand unanimous consensus method’s output tree. This would in particular implythat we could drop the condition that each cluster of every tree in P is compatiblewith all trees in P from Theorem 3.6. However, this stronger version of the theoremdoes not hold, as is demonstrated by Example 3.7. Example 3.7.
For the unrooted case, consider again Figure 1. Let P = ( T , T , T ),i.e. P employs two copies of T and one copy of T . For the rooted case, we introducea root on the inner edges of T and T , respectively. It can easily be seen that inboth cases, we have M R ( P ) = T , whereas ϕ ( P ) is the star tree (i.e. T in Figure1; in the rooted case the only inner vertex of T is then the root). So in particular,we have M R ( P ) ϕ ( P ). Since M R is unanimous (it is even regular, cf. [6]) andrefinement-stable by Theorem 3.1, this shows that the conditions of Theorem 3.6cannot be relaxed.However, as we will now show, if there is a cluster (or split, respectively) that isincompatible with a cluster (or split) that is compatible with all trees in a profile P , then this cluster cannot be contained in the output tree of any regular andrefinement-stable consensus method. Proposition 3.8.
Suppose φ is a regular, refinement-stable consensus method. Let P = ( T , . . . , T k ) be a profile of k rooted (or unrooted) phylogenetic X -trees, andlet c and c be two clusters (or σ and σ be two splits) such that c (or σ ) iscompatible with all trees in P , while c (or σ ) is not compatible with c (or σ ,respectively). Then, φ ( P ) does not contain c (or σ ).Proof. To see this, suppose φ ( P ) contained c (or σ ). Then we can refine all treesin P to contain c (or σ ), forming a new profile P ′ (this must be possible as c (or σ , respectively) is compatible with all trees in P by assumption). Then bydefinition of the strict consensus, Γ( P ′ ) contains c (or σ ). By Theorem 3.5, thisimplies φ ( P ′ ) contains c (or σ ). Thus, by refinement stability, φ ( P ) (cid:22) φ ( P ′ ). Butthis is impossible, since this would imply that φ ( P ′ ) contains both c and c (or σ and σ ), but these are incompatible by assumption. So this is a contradiction, whichshows that c (or σ ) cannot be contained in φ ( P ) to begin with. This completesthe proof.As we will now show, the previous proposition implies that if there are twoclusters (or splits) that are incompatible with one another, neither one of them canbe contained in the tree generated by a regular and refinement-stable consensusmethod, even if both of them are compatible with all input trees. Corollary 3.9.
Suppose φ is a regular, refinement-stable consensus method. Let P = ( T , . . . , T k ) be a profile of k rooted (or unrooted) phylogenetic X -trees, and let c and c be two clusters (or σ and σ be two splits) compatible with all clusters splits) of all trees in P , but not with each other. Then φ ( P ) contains neither c nor c (neither σ nor σ ).Proof. As c (or σ ) is compatible with all trees in P , by Proposition 3.8, φ ( P )cannot contain c (or σ ). Swapping the roles of c and c , however, as now they are both compatible with all trees in P , yields that by the same argument, φ ( P ) cannotcontain c (or σ ). This completes the proof.So just because two clusters (or splits) are incompatible with one another, theycannot be contained in the output of any regular and refinement-stable consensusmethod – even if both are compatible with all input trees. Again, note that byTheorem 3.1, this for instance applies to majority rule and strict consensus, i.e.to two of most frequently used consensus methods. Another direct consequence orProposition 3.8 is the following corollary. Corollary 3.10.
Suppose φ is a regular, refinement-stable consensus method. Let P = ( T , . . . , T k ) be a profile of k rooted (or unrooted) phylogenetic X -trees, and let T be some tree that is a refinement of all trees in P . Then φ ( P ) can consist only ofclusters (or splits, respectively) compatible with T .Proof. Let c be a cluster (or σ be a split) not compatible with T . Then c (or σ ) mustbe incompatible with at least one cluster e c ∈ C ∗ ( T ) (or split e σ ∈ Σ ∗ ( T )). Then,as T i (cid:22) T for all i = 1 , . . . , k by assumption, e c (or e σ ) is compatible with T i for all i = 1 , . . . , k . Thus, by Proposition 3.8, c (or σ ) cannot be contained in φ ( P ). Thiscompletes the proof.As we have seen in Theorem 3.1, majority rule and strict consensus are refinement-stable. Moreover, they are known to be regular [6]. We have also seen that not allmethods are refinement-stable, and also not all methods are regular – so is it possiblethat majority rule consensus and strict consensus are the only consensus methodsthat have both properties? This would be mathematically ‘nice’ because it wouldimply that these two properties already give a a full characterization for such meth-ods. However, as we will later on see in Theorem 3.12, this is unfortunately not thecase – in fact, there are even infinitely many such methods. As we will see lateron, the same is still true even if we enforce another biologically sensible property,namely non-contradiction, which we will consider in the following section. Before we turn our attention to the main result of this section, we will present afundamental insight into consensus methods that are both non-contradictory andrefinement-stable. In particular, we will show that under these circumstances, theoutput tree contains only clusters (or splits, respectively) that are already presentin the input profile. Note that this is not automatically the case: non-contradictionrequires only compatibility with at least one input tree, not containment.15 roposition 3.11.
Let φ be a refinement-stable and non-contradictory consensusmethod. Let P = ( T , . . . , T k ) be a profile of phylogenetic X -trees. Then for everycluster c (or split σ , respectively) in φ ( P ) , there is at least one tree T i that displays c (or σ ).Proof. Let T = φ ( P ) and assume T induces c (or σ ). Suppose, seeking a con-tradiction, that c (or σ ) is not induced by any tree in the profile P . First of all, | c | ≥ c would only contain one leaf and would thus be induced by allphylogenetic X -trees; the same applies to σ in the unrooted case).We now first consider the rooted case. Note that each T i must contain a subtree t i which contains all leaves of cluster c and such that t i is minimal with this property(for i = 1 , . . . , k ). Also note that as c is not induced by any T i ∈ P , the clusterinduced by t i , say c i , is not identical to c (for i = 1 , . . . , k ).For each T i , we now construct a refinement T ′ i which is incompatible with c .Therefore, we distinguish two cases for each T i : If c is already incompatible with T i ,we set T ′ i := T i . Else, if c is compatible with T i , we proceed as follows: In this case, t i contains at least one maximal pendant subtree t ∗ i (possibly with only one leaf)whose taxon set is disjoint from c . This is due to the fact that it must be possibleto introduce a single edge to separate c from all other leaves of t i (else, T i and c would not be compatible). Furthermore, it is important to note that the removal ofroot ρ i of subtree t i would then subdivide cluster c into at least two parts. This isdue to the fact that if c was a pendant cluster in t i , c would be induced by T i . Inturn, this means that in t i , there are at least two different maximal subtrees t i and t i which only lead to taxa in c (as T i is compatible with c ).Now, we refine each t i and thus also each T i to get T ′ i as follows: • In t i , we introduce an edge e i leading to the new cluster formed by t ∗ i and t i . • The new cluster consisting of the leaves of t ∗ i and t i will be called e c i . • The new tree, resulting from T i by introducing edge e i , will be called T ′ i .Now, note that c must be incompatible with e c i : Edge e i separates the taxa of t i from those of t i , which all belong to cluster c , and e c i contains taxa of t ∗ i which are not contained in c .Therefore, in summary, none of the T ′ i (neither the ones that were incompatiblewith c right away nor the ones which we generated by refining trees compatible with c appropriately) are compatible with c , but T ′ i is a refinement of T i for i = 1 , . . . , k .Now let P ′ = ( T ′ , . . . , T ′ k ). As φ is refinement-stable, T ′ := φ ( P ′ ) is a refinementof T and thus also necessarily induces c . However, as explained before, as c is notcompatible with any T ′ i (for i = 1 , . . . , k ), φ cannot be non-contradictory – otherwise,as c is incompatible with all trees in P ′ , c could not be contained in T ′ . So this isa contradiction. Therefore, the assumption was wrong and c must be displayed bysome tree in P .We now consider the unrooted case, in which T induces a split σ = A | B thatis not induced by any tree in P . We have already stated that | σ | ≥
2, where16 σ | = min {| A | , | B |} as defined in [11]. Then, there some T i ∈ P that are compatiblewith σ (this has to be the case as φ is non-contradictory), and possibly some thatare not. For now we disregard the ones that are incompatible with σ (if there areany).For all others, i.e. for all T i ∈ P that are compatible with σ , note that each such T i ∈ P contains only splits that are compatible with σ , but as σ is by assumptionnot induced by any of the T i , this implies that we could add σ to T i to get a tree T ′ i (which would be unique by Theorem 2.1). Note that this must mean that none ofthese T i are binary (otherwise we could not add an additional split), and thus eachsuch T i contains at least one vertex of degree at least 4. For σ = A | B , let us assumethat all leaves in T i that belong to A are colored red and all leaves of B are coloredgreen. In the following, we call a tree monochromatic if it only contains leaves ofone color.We now argue that T i must contain a vertex u of degree at least 4 such that theremoval of u would lead to only monochromatic subtrees, and in particular at leasttwo red ones and at least two green ones. Assume this is not the case: • First, assume that all vertices of degree at least 4 lead to at least one non-monochromatic subtree. Then for all such vertices v , it impossible to replacethem by two new vertices v and v connected by an edge e σ separating thered from the green subtrees, i.e. A from B . In other words, it is impossible toadd an edge that induces σ , which contradicts the compatibility of T i with σ . • So now that we know that there is a vertex u . of degree at least 4 whoseremoval leads to only monochromatic subtrees, assume that there are not atleast two subtrees of each color. In this case, if there is no, say, red subtree,all leaves are green. This contradicts the fact that | σ | ≥
2. So there has to beat least one subtree of each color. However, assume that there is only one redsubtree. Then the edge e leading from u to this subtree clearly separates allred leaves from all green leaves, in other words, e separates A from B . This isa contradiction, as we assume that no T i induces σ .So there must be a vertex u of degree at least 4 whose removal leads to onlymonochromatic subtrees, at least two of which, say T A and T A , contain only leavesfrom A and at least two of which, say T B and T B , contain only leaves from B .We now refine T i to give T ′ i as follows: • Introduce two new vertices u A and u B . • Delete u . • Connect all except for one of the red subtrees formerly incident to u now to u A by new edges. • Connect all except for one of the green subtrees formerly incident to u now to u B by new edges. 17 Connect the remaining red subtree to u B . • Connect the remaining green subtree to u A .Clearly, T ′ i refines T i : It contains all splits of T i , but it additionally contains asplit σ ′ = A ′ | B ′ , which is incompatible with σ , because we have that A ∩ A ′ , A ∩ B ′ , B ∩ A ′ and B ∩ B ′ are all non-empty by construction.So all T i in P that are compatible with σ have been refined to trees T ′ i which arenot compatible with σ . For the other trees T i in P which are already incompatiblewith σ , we set T ′ i := T i .This way, we can now consider profile P ′ := ( T ′ , . . . , T ′ k ), which only containstrees that are incompatible with σ . As φ is refinement-stable, T ′ := φ ( P ′ ) is arefinement of T and thus also necessarily induces σ . However, as explained before,as σ is not compatible with any T ′ i (for i = 1 , . . . , k ), φ cannot be non-contradictory– otherwise, as σ is incompatible with all trees in P ′ , σ could not be contained in T ′ . So this is a contradiction. Therefore, the assumption was wrong and σ must bedisplayed by some tree in P . This completes the proof.While we have already seen that M R and Γ are regular, refinement-stable andnon-contradictory (and all these are desirable properties of any consensus method,after all), whereas γ is not (as it lacks refinement-stability), the question remains ifthis combination makes Γ and M R unique or if there are other consensus methods –biologically justified or not – that also share all these properties. The following the-orem, which is the main result of this section, states that there are in fact infinitelymany such consensus methods, so these properties are not unique to the mentionedones.
Theorem 3.12.
There are infinitely many consensus methods that are regular,refinement-stable and non-contradictory.Proof.
For the rooted case, consider the profile P = ( T ′ , T ), where T and T arethe two phylogenetic X -trees depicted in Figure 3 and T ′ is a refinement of T (notnecessarily a strict one, so we may have T ′ = T ), and where X = { , . . . , n } . Forthe unrooted case, consider the exact same profile, but suppress the root, i.e. replacethe two edges incident to the degree-2 root vertex by a single edge.Note that the exact structure of the trees depends on whether n = 2 m + 2 or n = 2 m + 3 for some m ∈ N , i.e. whether n is even or odd. Note that T ∗ ...m and T ∗ m +1 ... m are identical if leaf labels are disregarded, i.e. they correspond to the sametreeshape, say T ∗ , but T ∗ can be chosen to be any rooted binary tree (note thatthis implies that there are W E ( m ) many choices for T ∗ , where W E ( m ) denotes the m th Wedderburn Etherington number [15, Sequence A001190]).We now define a consensus method φ as follows: If e P = P , i.e. if e P contains T ′ and T , then we define φ ( e P ) := T . Let π be a permutation on X . Let π ( T )denote a version of T in which the leaves are permuted according to π , where T isa phylogenetic X -tree. Then, we also define φ ( { π ( T ′ ) , π ( T } ) := π ( T ).18n all other cases, i.e. for all other input profiles e P , we define φ ( e P ) := Γ( e P ).We now argue that φ is regular, refinement-stable and non-contradictory. Una-nimity follows by the unanimity of Γ, and neutrality follows by Γ for all profiles thatare not e P , and for e P it follows by our consideration of permutation π . To checkanonymity, note that T ′ is either such that T ′ (cid:22) T or such that T and T havecompletely different tree shapes (if T gets refined in a way that does not form T ,namely by combining leaf 2 m +1 with one of the copies of T ∗ to form a new subtree).This shows that the only time when a permutation can modify T ′ to give T and T to give T ′ is when T ′ and T are isomorphic. However, in this case we actuallyhave by unanimity that φ returns precisely this tree, so for anonymity, there remainsnothing to show. Thus, φ is regular.To see that φ is actually refinement-stable, recall that Γ is refinement-stable byTheorem 3.1, and if a profile e P consists of π ( T ′ ) and π ( T ) for some permutation π , then φ returns π ( T ), whether T ′ equals T or is a strictly refined version of it.Thus, in summary, φ is refinement-stable.Last, we need to show that φ is non-contradictory. To see this, recall that byProposition 3.3, Γ is non-contradictory, and if a profile e P consists of π ( T ′ ) and π ( T ) for some permutation π , then φ returns π ( T ). So in particular, all splits of φ (( π ( T ′ ) , π ( T ))) = π ( T ) are not only compatible with at least one of the inputtrees, but even contained in it, as π ( T ) is contained in the input profile.So any consensus method φ of the type described here is regular, refinement-stable and non-contradictory. But we can construct infinitely many such methodsby varying the trees depicted in Figure 3: We can, for instance, consider differentvalues for n , so the cardinality of the set of such examples (and thus of such consensusmethods) equals the cardinality of N . This completes the proof. In this section, we take a closer look on MRP and MRC as consensus methods. Inparticular, we analyze them concerning refinement-stability and non-contradiction.We start with the first main theorem of this section.
Theorem 3.13.
MRP and MRC are not refinement-stable.Proof.
Consider again Figure 1. We consider profiles P = ( T , T , T ) and P =( T , T , T ). We consider the non-trivial splits induced by these profiles and codethem as binary characters as explained in Section 2.1. Then we can define B for P (and, analogously, B for P ) as in Section 2.3. Then, for P we have B = ∅ ∪ ∅ ∪ { } = { } , as the star tree T has no non-trivial splits. However, ifwe refine both star trees such that we derive P (note that T is a refinement of T ),we derive B = { , , } .We now start with considering MRP: It can be easily verified that the uniqueMRP tree for P is T : We have ps (1100 , T ) = ps (1100 , T ) = ps (1100 , T ) = 2,19 T ∗ ,...,m T ∗ m +1 ,..., m m leaves m leaves 2 m + 1 2 m + 2 2 m + 3 T T ∗ ,...,m T ∗ m +1 ,..., m m leaves m leaves 2 m + 1 2 m + 2 2 m + 3 Figure 3: Profile In the proof of Theorem 3.12, we have P = ( T ′ , T ) on n = 2 m + 2 or n = 2 m + 3leaves, respectively, where T ′ is a refinement of T as depicted here. T ∗ ...m and T ∗ m +1 ... m areboth trees with the same binary treeshape T ∗ , e.g. a so-called caterpillar (i.e. a rooted binarytree with precisely one cherry, i.e. one pair of leaves sharing the same parent), but the leaves of T ∗ ...m are bijectively labelled by { , . . . , m } and the leaves of T ∗ m +1 ... m are bijectively labelled by { m + 1 , . . . , m } . Note that the dashed parts of the trees only exist if n is odd. ps (1100 , T ) = 1, so in total ps ( B , T ) < ps ( B , T i ) for i = 1 , ,
4. However, as ps (1010 , T ) = 1 and ps (1010 , T ) = ps (1010 , T ) = ps (1010 , T ) = 2, we also havethat ps ( B , T ) = ps ( B , T ) = 2 + 2 + 2 = 6, ps ( B , T ) = 1 + 2 + 2 = 5 and ps ( B , T ) = 2 + 1 + 1 = 4. So in total, T is the unique MRP tree for P . Thisshows that the sets of MRP trees for P and P are disjoint, even though P is arefinement of P . Thus, MRP is not refinement-stable.Now let us consider MRC: In B , there is only one character, namely 1100, andthis character induces tree T . Thus, T is the unique MRC tree for P . On theother hand, in B = { , , } we have both 1100 and 1010, which areincompatible (as the splits 12 |
34 and 13 |
24 are incompatible). Thus, not all threecharacters of B can be compatible. The maximum number of compatible charactersin B is therefore 2. As no subset containing both 1100 and 1010 can be compatible,it turns out that the unique maximal subset of compatible characters is { , } ,and this subset induces tree T . Therefore, T is the unique MRC tree for P . Thisshows that the sets of MRC trees for P and P are disjoint, even though P is arefinement of P . Thus, MRC is not refinement-stable.This completes the proof.We will now turn our attention to non-contradiction, which shows a severe dif-ference between MRP and MRC, as one of these methods turns out to be non-contradictory, while the other one is not. We start with the following negativeresult concerning MRP. Theorem 3.14.
MRP is not non-contradictory.Proof.
Consider profile P = ( T , T ) consisting of the two trees T and T fromFigure 4. Moreover, consider the corresponding alignment of these two trees asgiven in Table 1. It can be easily seen that neither T nor T are MP trees for thisalignment; in fact, the alignment has parsimony score 13 on tree T from Figure 4,but 14 on both T and T , respectively (cf. Table 1). Moreover, we performed anexhaustive search using the computer algebra system Mathematica [16] in order toverify that T is even the unique MP tree for this alignment. However, T containsthe split σ = 1 , , , | , , T and T . As T and T are binary, this (by Theorem 2.1) necessarily means that σ is incompatible with both trees, because no more split can be added to said trees.This completes the proof.Next, we present a positive result on MRC, before we turn our attention to“future-proofing” MRP and MRC in the sense presented in [6], i.e. concerning theaddition of new taxa. Theorem 3.15.
MRC is non-contradictory.Proof.
By definition, the MRC tree is built by taking a maximum set of compatiblesplits of the input tree and combining them into a unique tree, e.g. using the so-called tree popping algorithm. This means that the MRC tree by definition only21
Figure 4: Trees T , T and T as needed for the proof of Theorem 3.14. The alignment for P = ( T , T ) is depicted in Table 1. alignment1 1 1 1 1 1 1 10 0 0 0 0 1 1 10 0 0 0 0 0 0 10 0 0 1 1 1 1 10 0 1 1 0 0 0 00 1 1 1 0 0 1 11 1 1 1 0 0 0 0 p s T t o t a l T T T and T into binarycharacters and its corresponding parsimony scores on T , T and T from Figure 4.22mploys splits that occur in at least one input tree. Note that this in particularimplies that MRC is non-contradictory. This completes the proof. In order to complement the results of [6], we briefly consider MRP and MRC inthe light of adding more taxa (rather than more splits or clusters). This scenario isfor instance relevant concerning the discovery of new species. The results stated inthe following and the construction of the particular examples were strongly inspiredby [10]. As stated above, both MRP and MRC are strictly speaking not consensusmethods as their output may be a set of trees rather than a single tree. However, wewill here focus on a profile P for which both MRP and MRC produce a unique tree,namely e P = ( e T , e T , e T , e T , e T , e T ), where e T , e T , e T are as depicted in Figure 5. Now,we add taxon 5 to all trees in e P such that we get profile P = ( T , T , T , T , T , T ),where again T , . . . , T are as depicted in Figure 5. Note that T \{ } = e T , T \{ } = T \ { } = T \ { } = e T , and T \ { } = e T . Translating all non-trivial splits of P and e P to characters leads to alignments A P and A e p as depicted in Figures 6 and 6.We used the computer algebra system Mathematica [16] to prove by exhaustivesearch that T is both the unique MRP and MRC tree for A P , and that e T is boththe unique MRP and MRC tree for A e P . However, as e T is not a subtree of T , thisshows that even in the case where MRP and MRC agree and additionally lead toa unique tree, additional taxa can change the predetermined relationships of thespecies in question. Thus, MRP and MRC are not ‘future-proof’ in the sense of [6]. We introduced two new properties which are desirable for sensible phylogenetic con-sensus methods, namely refinement-stability and non-contradiction. We analyzedseven established consensus methods and found that only two of them, namely strictconsensus and majority-rule consensus, are refinement-stable, whereas the other five(loose consensus, Adams consensus, Aho consensus, MRP and MRC) are not. Wealso proved that strict, majority-rule and loose consensus as well as MRC are allnon-contradictory, whereas Adams and Aho consensus as well as MRP are not.We also analyzed the implications that the properties of refinement-stability andnon-contradiction have on any refinement method, established or not, and surpris-ingly found that there are infinitely many consensus methods which are regular,refinement-stable and non-contradictory. As of the established consensus methodswe analyzed only two (namely strict and majority-rule) fit that description, it wouldbe interesting to see if there are other biologically plausible consensus methods whichhave these properties. This is a possible area for future investigations.23
CCAA
CCC CCAA AAA
14 23
CCAA
14 23
CCAACAACA
14 23
CCAA
AACCCCC CCAAAAAC CC CCAA AAA CC CA AA
AC AC
CCC CAA AA CA
45 132 ∅ T g T P g P g T T T T T g T Figure 5: Profile e P , which can be turned into profile P by adding information on taxon 5, leadingto alignments A e P and A P as depicted in Figures 6 and 7, respectively. T is the unique MRP andMRC tree for P , while f T is the unique MRP and MRC tree for e T . e P :
1: A A
A A A
2: A A
C C C
3: C C
C C C
4: C C
A A A
Figure 6: Alignment A e P , which has both a unique parsimony tree as well as a unique compatibilitytree: both of them equal f T as depicted in Figure 5. The unique maximum set of compatiblecharacters in A e P is hightlighted in bold. A P : A A A A
A A A A A A A A A A A
C A C C C C A C A C A
C C C C C A A C C C C
A A C A A A C C C C C
C C A A C C C Figure 7: Alignment A P , which has both a unique parsimony tree as well as a unique compatibilitytree: both of them equal T as depicted in Figure 5. The unique maximum set of compatiblecharacters in A P is highlighted in bold. Acknowledgement
MF wishes to thank the joint research project DIG-IT! supported by the EuropeanSocial Fund (ESF), reference: ESF/14-BM-A55- 0017/19, and the Ministry of Ed-ucation, Science and Culture of MecklenburgVorpommern, Germany. MH thanksProf. Dr. W. F. Martin, the Volkswagen Foundation 93 046 grant and the ERCAdvanced Grant No. 666053 for their support during this research.
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