inferring trees from trees consensus and supertree methods
DESCRIPTION
Department of Zoology, The Natural History Museum [email protected]. Inferring Trees from Trees Consensus and Supertree Methods. Mark Wilkinson. CONSENSUS “general or widespread agreement”. Consensus tree – a tree depicting agreement among a set of trees - a representation of a set of trees - PowerPoint PPT PresentationTRANSCRIPT
Department of Zoology,Department of Zoology,The Natural History MuseumThe Natural History [email protected]@bmnh.org
Inferring Trees from TreesInferring Trees from TreesConsensus and Supertree MethodsConsensus and Supertree Methods
Mark Mark WilkinsonWilkinson
CONSENSUSCONSENSUS“general or widespread “general or widespread
agreement”agreement” Consensus treeConsensus tree – a tree depicting – a tree depicting agreement among a set of treesagreement among a set of trees
- a representation of a set of trees- a representation of a set of trees- a phylogenetic inference from a set of - a phylogenetic inference from a set of
treestrees Consensus methodConsensus method – a technique for – a technique for
producing consensus trees (of a producing consensus trees (of a particular type)particular type)
Consensus indexConsensus index – a measure of the – a measure of the agreement among a set of trees (based agreement among a set of trees (based on their consensus tree)on their consensus tree)
Uses of Consensus TreesUses of Consensus Trees• Consensus trees are used to represent (or make Consensus trees are used to represent (or make
inferences from) multiple trees inferences from) multiple trees Agreement (conservative)Agreement (conservative)Central tendency (liberal)Central tendency (liberal)
• There are a number of different contexts in which There are a number of different contexts in which this may be of interest (sets of trees can be obtained this may be of interest (sets of trees can be obtained in a variety of ways)in a variety of ways)
• The ultimate aims may be quite differentThe ultimate aims may be quite different• Different methods may be more or less appropriate Different methods may be more or less appropriate
given the aim/contextgiven the aim/context
Mathematician’s Mathematician’s PerspectivePerspective
Subsequently there has been an amazing proliferation of consensus Subsequently there has been an amazing proliferation of consensus methods and consensus indices): a proliferation stimulated by methods and consensus indices): a proliferation stimulated by confusions, confusions, disagreements, and uncertaintiesdisagreements, and uncertainties concerning what consensus methods concerning what consensus methods depict and what consensus indices measure. Thus, for example, depict and what consensus indices measure. Thus, for example, consensus indices for trees are understood to measure agreement, consensus indices for trees are understood to measure agreement, balance, information, resolution, shape, similarity, and symmetry. One balance, information, resolution, shape, similarity, and symmetry. One has the impression that taxonomists do not know (or cannot agree on) has the impression that taxonomists do not know (or cannot agree on) what consensus objects should depict or how it should be depicted; they what consensus objects should depict or how it should be depicted; they do not know (or cannot agree on) what consensus indices should measure do not know (or cannot agree on) what consensus indices should measure or how it should be quantified. Consequently, taxonomists may not or how it should be quantified. Consequently, taxonomists may not appreciate (or do not articulate) the relationships that might or should appreciate (or do not articulate) the relationships that might or should exist between consensus method and consensus index.exist between consensus method and consensus index.
Day and McMorris 1985Day and McMorris 1985
Strict (Component) CMStrict (Component) CM• Uniquely defined in terms of two Uniquely defined in terms of two propertiesproperties• ParetoPareto - if a component is present in all - if a component is present in all the input trees it is in the consensusthe input trees it is in the consensus
• StrictStrict - if a component is in the consensus - if a component is in the consensus it is present in all the input treesit is present in all the input trees
• Can also be defined in terms of Can also be defined in terms of • an algorithman algorithm• an objective functionan objective function
Strict CMStrict CM(s)(s)• Require complete agreement across all the input Require complete agreement across all the input
trees and show relationships that would be true if trees and show relationships that would be true if any input tree were trueany input tree were true
• With MPTs they show only those relationships that With MPTs they show only those relationships that are unambiguously supported by the parsimonious are unambiguously supported by the parsimonious interpretation of the datainterpretation of the data
• The commonest method focuses on components The commonest method focuses on components (clusters, groups, splits, clades or monophyletic (clusters, groups, splits, clades or monophyletic groups)groups)
• This method produces a consensus tree that This method produces a consensus tree that includes all (includes all (ParetoPareto) and only () and only (strictstrict) the common ) the common clades Other relationships (those in which the input clades Other relationships (those in which the input trees disagree) are shown as unresolved polytomiestrees disagree) are shown as unresolved polytomies
• Component version widely implementedComponent version widely implemented
Strict CMStrict CM(s)(s)
A B C D E F G A B C E D F G
TWO INPUT TREES
A B C D E F G
STRICT (COMPONENT) CONSENSUS TREE
EB C A D
EA B C D
EA C B D
EA C B D
Interpreting PolytomiesInterpreting Polytomies• Polytomies in trees have alternative interpretations.Polytomies in trees have alternative interpretations.• The The hardhard interpretation interpretation ‘‘multiple speciation’multiple speciation’• The The softsoft interpretation interpretation
‘‘uncertain resolution’uncertain resolution’• The soft interpretation The soft interpretation
is appropriate for strict is appropriate for strict component consensus trees component consensus trees
• The consensus permits allThe consensus permits allresolutions of the polytomy resolutions of the polytomy
(i.e. it does not conflict with any resolution)(i.e. it does not conflict with any resolution)
Semi-strict CMSemi-strict CM(s)(s) • Semi-strict methods require assertion of a Semi-strict methods require assertion of a
relationship by one or more trees and non-relationship by one or more trees and non-contradiction by any treecontradiction by any tree
• The commonest method focuses on The commonest method focuses on components/cladescomponents/clades
• Produces a tree including all components that are Produces a tree including all components that are present and uncontradicted in the input trees - all present and uncontradicted in the input trees - all that could be true if any input tree were truethat could be true if any input tree were true
• Generally, similar to the strict but may be more Generally, similar to the strict but may be more resolved when the input trees include polytomiesresolved when the input trees include polytomies
• It is based on the soft interpretation of polytomies It is based on the soft interpretation of polytomies • Other relationships are shown as unresolved Other relationships are shown as unresolved
polytomiespolytomies• Component version implemented in e.g. PAUPComponent version implemented in e.g. PAUP
Semi-strict (component) CMSemi-strict (component) CM TWO INPUT TREES
A B C D E F D E F A B C
D E FA B C
SEMI-STRICT (COMPONENT) CONSENSUS
Properties of Semi-strict Properties of Semi-strict CMCM(s)(s)
• Tend to produce more resolved consensus treesTend to produce more resolved consensus trees• Reasonable when combining trees based on Reasonable when combining trees based on
different data setdifferent data set• With trees based on a single data set extra With trees based on a single data set extra
resolution is of relationships that are not true of all resolution is of relationships that are not true of all the optimal trees:the optimal trees:the consensus includes relationships that are not supported the consensus includes relationships that are not supported
by all best interpretations of the data by all best interpretations of the data It may include relationships that cannot be simultaneously It may include relationships that cannot be simultaneously
supported by any parsimonious (or other) interpretation of supported by any parsimonious (or other) interpretation of the datathe data
These relationships might reasonably be considered less well These relationships might reasonably be considered less well supported (if supported at all)supported (if supported at all)
Semi-strict (component) CMSemi-strict (component) CMTWO INPUT TREES
SEMI-STRICT (component) CONSENSUS TREE
Equally Optimal TreesEqually Optimal Trees• Many phylogenetic analyses yield multiple equally Many phylogenetic analyses yield multiple equally
optimal treesoptimal trees• Multiple trees are due to either:Multiple trees are due to either:
Alternative equally optimal interpretations of conflicting dataAlternative equally optimal interpretations of conflicting dataMissing dataMissing dataOr bothOr both
• We can further select among these trees with We can further select among these trees with additional (secondary) criteria, butadditional (secondary) criteria, but
• A consensus tree may be needed to represent or draw A consensus tree may be needed to represent or draw conclusions about the set of MTPsconclusions about the set of MTPs
• Typically phylogeneticists are interested in Typically phylogeneticists are interested in relationships common to all the optimal trees relationships common to all the optimal trees (they (they want to know that a relationship in the consensus is want to know that a relationship in the consensus is in all the trees - in all the trees - STRICTSTRICT))
Loss of ResolutionLoss of Resolution• Generally, as the number of optimal trees increases Generally, as the number of optimal trees increases
the resolution of ‘strict consensus trees’ decreases. the resolution of ‘strict consensus trees’ decreases. • In the extreme, the ‘strict consensus tree’ may be In the extreme, the ‘strict consensus tree’ may be
completely unresolved/uninformative.completely unresolved/uninformative.• This extreme is sometimes met in practice (This extreme is sometimes met in practice (e.ge.g. .
fossilsfossils). ). • The ‘consensus tree’ can also be poorly resolved The ‘consensus tree’ can also be poorly resolved
when there are few optimal trees, and.furthermore.when there are few optimal trees, and.furthermore.• The optimal trees need not differ greatly, thus...The optimal trees need not differ greatly, thus...• Lack of resolution in a ‘strict consensus tree’ is not Lack of resolution in a ‘strict consensus tree’ is not
always a good guide to the level of agreement among always a good guide to the level of agreement among the optimal trees.the optimal trees.
Optimal trees need not Optimal trees need not differ greatly for ‘the differ greatly for ‘the consensus tree’ to be consensus tree’ to be
unresolvedunresolvedA B C D E F G A B C DE F G
Adams-2 ConsensusAdams-2 ConsensusAdams (1972) method was defined by a recursive algorithm, which, beginning at the root, identifies common sub-clusters using intersection rules(product partitions)
A B C F E D A CB F E D
INPUT TREES
A C B E F D
CONSENSUS
The basal splits in these trees yield the four clusters ABC, EFD, AC & BEFD. Their intersections yield AC, B & EFD and these produce the three branches at the base of the consensus tree. The procedure is repeated for the subtrees induced by E, F and D, which in this case are identical.
Adams Consensus and Adams Consensus and NestingNesting• Adams (1972) described the first consensus Adams (1972) described the first consensus
methods, only one of his methods is usedmethods, only one of his methods is used• Adams (1986) characterised his method in terms ofAdams (1986) characterised his method in terms of
nestingsnestings• A group X (e.g. AB) nests within another Y (e.g. A-D) A group X (e.g. AB) nests within another Y (e.g. A-D)
if the last common ancestor of Y is an ancestor of if the last common ancestor of Y is an ancestor of the last common ancestor of X the last common ancestor of X
• The Adams consensus tree includes all those The Adams consensus tree includes all those nestings that are in all the input trees nestings that are in all the input trees ((ParetoPareto) ) and and for all clusters displayed by the Adams tree there is for all clusters displayed by the Adams tree there is a corresponding nesting in each input tree (a corresponding nesting in each input tree (strictstrict).).
• They show only those nestings that are They show only those nestings that are unambiguously supported by the parsimonious unambiguously supported by the parsimonious interpretation of the datainterpretation of the data
• Implemented in e.g. PAUPImplemented in e.g. PAUP
Adams Consensus and Adams Consensus and NestingNesting
(1) taxa A & C are more closely related to each other than either is to taxa D, E, or F;
(2) taxa E & F are more closely related to each other than either is to taxa A, C, or D;
(3) taxon D is more closely related to E & F than it is to either A or C.
Swofford (1991)
But - Not quite right!
A B C F E D A CB F E D
INPUT TREES
A C B E F D
CONSENSUS
Adams ConsensusAdams Consensus(1) taxa A & C are more closely related to each other than either is to taxa D, E, or F;In each tree A & C are more closely related to each other than they are to D-F and/or B(AC)D-F and/or (AC)B
(2) taxa E & F are more closely related to each other than either is to taxa A, C, or D;(EF)D
(3) taxon D is more closely related to E & F than it is to either A or C.(D-F)AC and/or (D-F)B
D BA E C F
A B C F E D A CB F E D
INPUT TREES
A C B E F D
CONSENSUS
Adams polytomies are Adams polytomies are cladistically ambiguouscladistically ambiguous
What can be inferred from the consensus?(A,B)CD - No(AB)C - No(AB)D - No(AB)C and/or (AB)D
INPUT TREES
A B C D A D B C
A B C D
Note on the meaning of Note on the meaning of cladistic relationshipcladistic relationship
• Cladistic relationships are based on Cladistic relationships are based on recency of common ancestry (& recency of common ancestry (& dependent on rooted trees).dependent on rooted trees).
• Two taxa are more closely related to each Two taxa are more closely related to each other than either is to a third other than either is to a third iffiff they they share a more recent common ancestor - share a more recent common ancestor - e.g. (AB)CDE.e.g. (AB)CDE.
• Nestings are also based on common Nestings are also based on common ancestry but ancestry but nestings are ambiguousnestings are ambiguous with with respect to cladistic relationships - e.g. respect to cladistic relationships - e.g. {AB}CD = (AB)C and/or (AB)D{AB}CD = (AB)C and/or (AB)D
Properties of Adams Properties of Adams Consensus TreesConsensus Trees
• Adams consensus trees are more Adams consensus trees are more topologically sensitive to shared structure topologically sensitive to shared structure in input trees than is the strict component in input trees than is the strict component consensus, but...consensus, but...
• Care must be taken in the interpretation of Care must be taken in the interpretation of their ‘elastic’ polytomiestheir ‘elastic’ polytomies
• Adams consensus trees can include Adams consensus trees can include groupsgroups that don’t occur in any input tree (that don’t occur in any input tree (Rholf Rholf GroupsGroups))
• It exists only for rooted trees It exists only for rooted trees
Greatest Agreement Greatest Agreement SubtreesSubtrees
A B C D E F G
TWO INPUT TREES
GAS/LCP TREE
Taxon G is excluded
A G B C D E F
A B C D E FA B C D E F G
Strict component consensuscompletely unresolved
Strict Reduced CMStrict Reduced CM
A B C D E F G
TWO INPUT TREES
STRICT REDUCED CONSENSUS TREE
Taxon G is excluded
A G B C D E F
A B C D E F
A B C D E F G
Strict component consensus
B C D E F A C D E F
A B D E F
Agreement Subtrees
RhynchosaursH. gordoniH. huxleyiS. fischeriS. sanjaunensisSupradapedonNova ScotiaTexasIsalorhnchusAcrodentusR. spenceriR. brodeiR. articepsMesodapedonStenaulorhynchusHowesiaMesosuchus
1
H. gordoniH. huxleyiS. fischeriS. sanjaunensisSupradapedonNova ScotiaIsalorhnchusR. spenceriR. brodeiR. articepsMesodapedonStenaulorhynchusHowesiaMesosuchus
2
H. gordoniH. huxleyiS. fischeriS. sanjaunensisSupradapedonNova ScotiaIsalorhnchusR. spenceriR. brodeiR. articepsStenaulorhynchusHowesiaMesosuchus
3
H. gordoniH. huxleyiS. fischeriS. sanjaunensisNova ScotiaR. spenceriR. brodeiR. articeps
StenaulorhynchusHowesiaMesosuchus
4
Mesodapedon
H. gordoniH. huxleyiS. fischeri
R. spenceriR. brodeiR. articeps
StenaulorhynchusHowesiaMesosuchus
5
Mesodapedon
S. sanjaunensis
H. gordoniH. huxleyiS. fischeri
R. spenceriR. brodeiR. articepsStenaulorhynchusHowesiaMesosuchus
6
S. sanjaunensis
Fossil & Recent Arthropods
Fossil & Recent Arthropods
Majority-rule CMMajority-rule CM(s)(s) • Majority-rule consensus methods require agreement Majority-rule consensus methods require agreement
across a majority of the input treesacross a majority of the input trees• The commonest method focuses on The commonest method focuses on
components/cladescomponents/clades• This method produces a consensus tree that This method produces a consensus tree that
includes all and only those clades found in a includes all and only those clades found in a majority (>50%) of the input trees majority (>50%) of the input trees
• Majority components which are necessarily mutually Majority components which are necessarily mutually compatiblecompatible
• Other relationships are shown as unresolved Other relationships are shown as unresolved polytomiespolytomies
• Of particular use in bootstrapping, jackknifing, Of particular use in bootstrapping, jackknifing, quartet puzzling, Bayesian inference (quartet puzzling, Bayesian inference (with e.g. with e.g. average branch lengthsaverage branch lengths).).
• Component version widely implementedComponent version widely implemented
Majority-rule (component) Majority-rule (component) CMCM
A B C D E F G A B C E D F G
A B C E D F G
MAJORITY-RULE (COMPONENT) CONSENSUS
A B C E F D G
10066
66
66
66
THREE INPUT TREES
Numbers indicate frequencyof clades in the input trees
Properties of Majority-ruleProperties of Majority-rule
• Tend to produce more resolved consensus treesTend to produce more resolved consensus trees• Extra resolution is of relationships that are not true Extra resolution is of relationships that are not true
of all the optimal treesof all the optimal trees• In the context of equally optimal trees, this means In the context of equally optimal trees, this means
the consensus includes relationships that are not the consensus includes relationships that are not supported by all the best interpretations of the data supported by all the best interpretations of the data
• These relationships might reasonably be considered These relationships might reasonably be considered less well supported (if supported at all)less well supported (if supported at all)
• Related to the Related to the Median consensusMedian consensus (objective function (objective function minimises the sum of the symmetric differences minimises the sum of the symmetric differences between the consensus and input trees)between the consensus and input trees)
Adding minority componentsAdding minority components
• Further resolution can sometimes be Further resolution can sometimes be achieved by adding relationships that achieved by adding relationships that occur in a minority of trees.occur in a minority of trees.
• These must be compatible with the These must be compatible with the majority relationshipsmajority relationships
• Two approachesTwo approachesGreedy (PAUP)Greedy (PAUP)Frequency-difference (TNT)Frequency-difference (TNT)
Majority-ruleMajority-rule
A B C D E A B C D E
55%
X
Other Consensus Other Consensus methodsmethods
• A variety of other consensus methods A variety of other consensus methods have been devised but few implementedhave been devised but few implemented
• These include:These include:Other intersection rules based on cluster Other intersection rules based on cluster
heightheight
Nelson, Asymmetric Median and other clique Nelson, Asymmetric Median and other clique consensus methodsconsensus methods
Other matrix respresentation methods, e.g.Other matrix respresentation methods, e.g.
MRPMRP
Average consensusAverage consensus
A Consensus A Consensus ClassificationClassification
• Consensus trees vary with respect to:Consensus trees vary with respect to:The kind of agreement (components, triplets, The kind of agreement (components, triplets, nestings, subtrees)nestings, subtrees)
The level of agreement (strict, semi-strict, The level of agreement (strict, semi-strict, majority-rule, largest clique)majority-rule, largest clique)
AdamsAdams Reduced LCP / GAS Reduced LCP / GAS Full SplitsFull Splits NestingsNestings Splits Splits Subtrees Subtrees
StrictStrict Yes Yes Yes Yes Yes Yes YesYesSemi-strictSemi-strict YesYes ? ? Yes Yes ? ?Majority-ruleMajority-rule YesYes ? ? Yes Yes ? ?Nelson (clique)Nelson (clique) YesYes ? ? Yes Yes ? ?
Consensus methodsConsensus methods• Use strict methods to identify those relationships Use strict methods to identify those relationships
unambiguously supported by parsimonious unambiguously supported by parsimonious interpretation of the datainterpretation of the data
• Use more liberal (semi-strict, majority-rule) consensus Use more liberal (semi-strict, majority-rule) consensus methods for taxonomic congruencemethods for taxonomic congruence
• Use majority-rule methods in bootstrapping etc.Use majority-rule methods in bootstrapping etc.• Use Adams consensus when strict component Use Adams consensus when strict component
consensus is poorly resolved - if Adams is better consensus is poorly resolved - if Adams is better resolved use strict reduced consensusresolved use strict reduced consensus
• Use reduced methods where consensus trees are Use reduced methods where consensus trees are poorly resolvedpoorly resolved
• Avoid over-interpreting results from methods which Avoid over-interpreting results from methods which have ambiguous interpretationshave ambiguous interpretations
Gbrev
Gram
Pcoop
Galt
GlarvGsech
Hrost
GaltGlarv
Gsech
Pcoop
Gram
Gbrev
Hrost Hrost
Gram
Pcoop
Galt
GlarvGsech
Gbrev
Galt
Gram
Pcoop
Gbrev
HrostGsech
Glarv
Hrost
Gram
Pcoop
Gbrev
GlarvGsech
Galt
Hrost
Gram
Pcoop
Gbrev
GlarvGsech
Galt
Hrost
Gram
Pcoop
Gbrev
GlarvGsech
Galt
Input Trees
Consensus Trees
More or lessConservative
More or lessLiberal
GbrevPcoop
Galt
Glarv
Galt
Pcoop
Gram
Gbrev
Gram
Galt
GlarvGsech
Pcoop
Gbrev
HrostGsech
Hrost
Gbrev
Glarv
Galt
Hrost
Gram
Pcoop
Gbrev
GlarvGsech
Hrost
Gram
Pcoop
Gbrev
GlarvGsech
GaltGalt
Input Trees
More or lessConservative
More or lessLiberal
SuperTrees
Biologists want (Big) Biologists want (Big) TreesTrees
• ““Nothing in Biology makes sense except in the Nothing in Biology makes sense except in the light of evolution” Dobzhansky, 1973light of evolution” Dobzhansky, 1973
• The Tree of Life: Holy Grail of SystematicsThe Tree of Life: Holy Grail of Systematics• Bigger Trees: more powerful comparative Bigger Trees: more powerful comparative
analysesanalyses– AdaptationAdaptation– BiogeographyBiogeography– Speciation and diversificationSpeciation and diversification– ConservationConservation
CACD+BA=ACDB
When Input Trees ConflictWhen Input Trees Conflict• Semi-strictSemi-strict• Gene Tree ParsimonyGene Tree Parsimony • MinCut (modified Aho)MinCut (modified Aho)• Quartet puzzlingQuartet puzzling• Matrix representationsMatrix representations
• Splits (standard MRP, MRC, MRF)Splits (standard MRP, MRC, MRF)• Sister groups (Purvis MRP)Sister groups (Purvis MRP)• TripletsTriplets• QuartetsQuartets• Distances (MRD)Distances (MRD)
– Analysed withAnalysed with• Parsimony (MRP) - Parsimony (MRP) - , , • Clique (MRC)Clique (MRC)• MinFlip (MRF)MinFlip (MRF)• Least squaresLeast squares
A B C D
MRPMRP
Polycladida
FecampiidaTricladidaLecithoepitheliataKalyptorhyncha
FecampiidaNeodermataKalyptorhynchaTricladidaLecithoepitheliata
Tree 1 Tree 2
Fecampiida 111 111 111 111 1??11??111Neodermata ??? 111 ??? 111 ??????????Tricladida 111 100 111 10? ?1???11111Lecithoepitheliata 110 000 110 0?? ??111110??Polycladida 000 ??? 00? ??? 0000?0??0?Kalytporhyncha 100 110 1?? 110 111?0?0??0MRP-outgroup 000 000 000 000 0000000000
Component Purvis Triplet
MRPMRP
Fecampiida
Neodermata
Fecampiida
Neodermata
Fecampiida
Neodermata
TricladidaLecithoepitheliata
TricladidaLecithoepitheliata
Kalyptorhyncha
Polycladida
Polycladida Polycladida
Kalyptorhyncha
Kalyptorhyncha
Lecithoepitheliata
Tricladida
A
C
B
Components unordered - A,B,C irreversible - CTriplets - AQuartets - A & C
MRP – an unusual MRP – an unusual consensusconsensus
MRP, total evidence MRP, total evidence and taxonomic and taxonomic
congruencecongruence one twoA 0 1 1 0 0 0 0 A 0 1 1 0 0 0 0B 1 0 0 1 0 0 0 B 1 0 0 1 1 1 1C 1 1 1 0 1 1 1 C 1 1 1 0 1 1 1D 1 0 0 1 1 1 1 D 1 0 0 1 0 0 0
A B C D A D B CA B C D
A C B D
one* two*A 0 0 A 0 0B 1 0 B 1 1C 1 1 C 1 1D 1 1 D 1 0
Pisani & Wilkinson Fig. 1
CONSENSUS
MATRIX REPRESENTATIONS
CONSENSUSSUPERTREE
TOTAL EVIDENCE
SEPARATEANALYSIS
SEPARATEANALYSIS
TAXONOMIC CONGRUENCE
Majority-Majority-rulerule
D E A C D E B C D A B C
A B C D E A B C D E
B C D E A C D E A B
1 2 3 4
7 = 1 + 2 + 3
5 = 2 + 3 + 4 6 = 1 + 3 + 4
8 = 1 + 2 + 4
INPUT TREES
Goloboff and Pol (2002), Goloboff (2006)
Majority rule supertrees desirable in principle
Fundamental problem in generalising frequency of occurrence of groups
MRP is a (poor) surrogate
Majority-ruleMajority-rule• Majority-rule consensus is also a median Majority-rule consensus is also a median
tree for the symmetric differencetree for the symmetric difference
• Alternative basis for generalising Alternative basis for generalising beyond consensusbeyond consensus
• How to compute symmetric difference How to compute symmetric difference for trees with different leaf sets?for trees with different leaf sets?
• Convert into trees with identical leaf Convert into trees with identical leaf setssets• Prune leaves from supertreePrune leaves from supertree• Graft leaves onto supertreeGraft leaves onto supertree
ML SupertreesML Supertrees
‘‘Taxonomic CongruenceTaxonomic Congruence’’
‘‘Taxonomic CongruenceTaxonomic Congruence’’