1. Inferring Trees from Trees Consensus and Supertree Methods
Mark Wilkinson
Department of Zoology,
The Natural History Museum

2. CONSENSUS “general or widespread agreement”
Consensus tree – a tree depicting agreement among a set of trees
- a representation of a set of trees
- a phylogenetic inference from a set of trees
Consensus method – a technique for producing consensus trees (of a particular type)
Consensus index – a measure of the agreement among a set of trees (based on their consensus tree)

3. Uses of Consensus Trees
Consensus trees are used to represent (or make inferences from) multiple trees
Agreement (conservative)
Central tendency (liberal)
There are a number of different contexts in which this may be of interest (sets of trees can be obtained in a variety of ways)
The ultimate aims may be quite different
Different methods may be more or less appropriate given the aim/context

4. Mathematician’s Perspective
Subsequently there has been an amazing proliferation of consensus methods and consensus indices): a proliferation stimulated by confusions, disagreements, and uncertainties concerning what consensus methods depict and what consensus indices measure. Thus, for example, consensus indices for trees are understood to measure agreement, balance, information, resolution, shape, similarity, and symmetry. One has the impression that taxonomists do not know (or cannot agree on) what consensus objects should depict or how it should be depicted; they do not know (or cannot agree on) what consensus indices should measure or how it should be quantified. Consequently, taxonomists may not appreciate (or do not articulate) the relationships that might or should exist between consensus method and consensus index.
Day and McMorris 1985

5. Strict (Component) CM Uniquely defined in terms of two properties
Pareto - if a component is present in all the input trees it is in the consensus
Strict - if a component is in the consensus it is present in all the input trees
Can also be defined in terms of
an algorithm
an objective function

6. Strict CM(s)
Require complete agreement across all the input trees and show relationships that would be true if any input tree were true
With MPTs they show only those relationships that are unambiguously supported by the parsimonious interpretation of the data
The commonest method focuses on components (clusters, groups, splits, clades or monophyletic groups)
This method produces a consensus tree that includes all (Pareto) and only (strict) the common clades Other relationships (those in which the input trees disagree) are shown as unresolved polytomies
Component version widely implemented

7. Strict CM(s) TWO INPUT TREES STRICT (COMPONENT) CONSENSUS TREE A B C DF G A B C E D F G A B C D E F G
STRICT (COMPONENT) CONSENSUS TREE

8. Interpreting Polytomies
Polytomies in trees have alternative interpretations.
The hard interpretation
‘multiple speciation’
The soft interpretation
‘uncertain resolution’
is appropriate for strict
component consensus trees
The consensus permits all
resolutions of the polytomy
(i.e. it does not conflict with any resolution)

9. Semi-strict CM(s)
Semi-strict methods require assertion of a relationship by one or more trees and non-contradiction by any tree
The commonest method focuses on components/clades
Produces a tree including all components that are present and uncontradicted in the input trees - all that could be true if any input tree were true
Generally, similar to the strict but may be more resolved when the input trees include polytomies
It is based on the soft interpretation of polytomies
Other relationships are shown as unresolved polytomies
Component version implemented in e.g. PAUP

10. Semi-strict (component) CM
TWO INPUT TREES

11. Properties of Semi-strict CM(s)
Tend to produce more resolved consensus trees
Reasonable when combining trees based on different data set
With trees based on a single data set extra resolution is of relationships that are not true of all the optimal trees:
the consensus includes relationships that are not supported by all best interpretations of the data
It may include relationships that cannot be simultaneously supported by any parsimonious (or other) interpretation of the data
These relationships might reasonably be considered less well supported (if supported at all)

12. Semi-strict (component) CM
TWO INPUT TREES
SEMI-STRICT (component) CONSENSUS TREE

13. Equally Optimal Trees
Many phylogenetic analyses yield multiple equally optimal trees
Multiple trees are due to either:
Alternative equally optimal interpretations of conflicting data
Missing data
Or both
We can further select among these trees with additional (secondary) criteria, but
A consensus tree may be needed to represent or draw conclusions about the set of MTPs
Typically phylogeneticists are interested in relationships common to all the optimal trees (they want to know that a relationship in the consensus is in all the trees - STRICT)

14. Loss of Resolution
Generally, as the number of optimal trees increases the resolution of ‘strict consensus trees’ decreases.
In the extreme, the ‘strict consensus tree’ may be completely unresolved/uninformative.
This extreme is sometimes met in practice (e.g. fossils).
The ‘consensus tree’ can also be poorly resolved when there are few optimal trees, and.furthermore.
The optimal trees need not differ greatly, thus...
Lack of resolution in a ‘strict consensus tree’ is not always a good guide to the level of agreement among the optimal trees.

15. Optimal trees need not differ greatly for ‘the consensus tree’ to be unresolved

16. Adams-2 Consensus
Adams (1972) method was defined by a recursive algorithm, which, beginning at the root, identifies common sub-clusters using intersection rules
(product partitions)
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.

17. Adams Consensus and Nesting
Adams (1972) described the first consensus methods, only one of his methods is used
Adams (1986) characterised his method in terms of nestings
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 the last common ancestor of X
The Adams consensus tree includes all those nestings that are in all the input trees (Pareto) and for all clusters displayed by the Adams tree there is a corresponding nesting in each input tree (strict).
They show only those nestings that are unambiguously supported by the parsimonious interpretation of the data
Implemented in e.g. PAUP

18. Adams Consensus and Nesting
(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!

19. Adams 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

20. Adams polytomies are cladistically 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

21. Note on the meaning of cladistic relationship
Cladistic relationships are based on recency of common ancestry (& dependent on rooted trees).
Two taxa are more closely related to each other than either is to a third iff they share a more recent common ancestor - e.g. (AB)CDE.
Nestings are also based on common ancestry but nestings are ambiguous with respect to cladistic relationships - e.g. {AB}CD = (AB)C and/or (AB)D

22. Properties of Adams Consensus Trees
Adams consensus trees are more topologically sensitive to shared structure in input trees than is the strict component consensus, but...
Care must be taken in the interpretation of their ‘elastic’ polytomies
Adams consensus trees can include groups that don’t occur in any input tree (Rholf Groups)
It exists only for rooted trees

23. Greatest Agreement Subtrees
TWO INPUT TREES A B C D E F G A G B C D E F A B C D E F G A B C D E F
Strict component consensus
completely unresolved
GAS/LCP TREE
Taxon G is excluded

24. Strict Reduced CM Strict component consensus TWO INPUT TREESA B C D E F G A G B C D E F A B C D E F G
Strict component
consensus A B C D E F B C D E F A
STRICT REDUCED
CONSENSUS TREE
Agreement
Subtrees
Taxon G is excluded

25. Rhynchosaurs

26. Fossil & Recent Arthropods

27. Fossil & Recent Arthropods

28. Majority-rule CM(s)
Majority-rule consensus methods require agreement across a majority of the input trees
The commonest method focuses on components/clades
This method produces a consensus tree that includes all and only those clades found in a majority (>50%) of the input trees
Majority components which are necessarily mutually compatible
Other relationships are shown as unresolved polytomies
Of particular use in bootstrapping, jackknifing, quartet puzzling, Bayesian inference (with e.g. average branch lengths).
Component version widely implemented

29. Majority-rule (component) CM
THREE INPUT TREES A B C D E F G A B C E F D G A B C E D F G A B C E D F G
100 66
Numbers indicate frequency
of clades in the input trees 66 66 66
MAJORITY-RULE (COMPONENT) CONSENSUS

30. Properties of Majority-rule
Tend to produce more resolved consensus trees
Extra resolution is of relationships that are not true of all the optimal trees
In the context of equally optimal trees, this means the consensus includes relationships that are not supported by all the best interpretations of the data
These relationships might reasonably be considered less well supported (if supported at all)
Related to the Median consensus (objective function minimises the sum of the symmetric differences between the consensus and input trees)

31. Adding minority components
Further resolution can sometimes be achieved by adding relationships that occur in a minority of trees.
These must be compatible with the majority relationships
Two approaches
Greedy (PAUP)
Frequency-difference (TNT)

32. Majority-rule

33. Other Consensus methods
A variety of other consensus methods have been devised but few implemented
These include:
Other intersection rules based on cluster height
Nelson, Asymmetric Median and other clique consensus methods
Other matrix respresentation methods, e.g.
MRP
Average consensus

34. A Consensus Classification
Consensus trees vary with respect to:
The kind of agreement (components, triplets, nestings, subtrees)
The level of agreement (strict, semi-strict, majority-rule, largest clique)
Adams Reduced LCP / GAS
Full Splits Nestings Splits Subtrees
Strict Yes Yes Yes Yes
Semi-strict Yes ? Yes ?
Majority-rule Yes ? Yes ?
Nelson (clique) Yes ? Yes ?

35. Consensus methods
Use strict methods to identify those relationships unambiguously supported by parsimonious interpretation of the data
Use more liberal (semi-strict, majority-rule) consensus methods for taxonomic congruence
Use majority-rule methods in bootstrapping etc.
Use Adams consensus when strict component consensus is poorly resolved - if Adams is better resolved use strict reduced consensus
Use reduced methods where consensus trees are poorly resolved
Avoid over-interpreting results from methods which have ambiguous interpretations

36. Input Trees Consensus Trees More or less Conservative More or less
Liberal

37. Input Trees
SuperTrees
More or less
Conservative
More or less
Liberal

38. Biologists want (Big) Trees
“Nothing in Biology makes sense except in the light of evolution” Dobzhansky, 1973
The Tree of Life: Holy Grail of Systematics
Bigger Trees: more powerful comparative analyses
Adaptation
Biogeography
Speciation and diversification
Conservation

40. When Input Trees Conflict
Semi-strict
Gene Tree Parsimony
MinCut (modified Aho)
Quartet puzzling
Matrix representations
Splits (standard MRP, MRC, MRF)
Sister groups (Purvis MRP)
Triplets
Quartets
Distances (MRD)
Analysed with
Parsimony (MRP) - , 
Clique (MRC)
MinFlip (MRF)
Least squares

41. MRP
Tree 1
Tree 2
Component Purvis Triplet
Fecampiida ??11??111
Neodermata ??? 111 ??? 111 ??????????
Tricladida ? ?1???11111
Lecithoepitheliata ?? ??111110??
Polycladida 000 ??? 00? ??? 0000?0??0?
Kalytporhyncha ?? ?0?0??0
MRP-outgroup

42. MRP Components unordered - A,B,C irreversible - C Triplets - A
Quartets - A & C

43. MRP – an unusual consensus

44. MRP, total evidence and taxonomic congruence

45. Majority-rule 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

46. Majority-rule
Majority-rule consensus is also a median tree for the symmetric difference
Alternative basis for generalising beyond consensus
How to compute symmetric difference for trees with different leaf sets?
Convert into trees with identical leaf sets
Prune leaves from supertree
Graft leaves onto supertree

47. ML Supertrees

48. ‘Taxonomic Congruence’

49. ‘Taxonomic Congruence’