Department of Zoology,Department of Zoology,The Natural History MuseumThe Natural History Museummw@bmnh.orgmw@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’’