GENERALISING FROM CONSENSUS TO SUPERTREES

Mark Wilkinson Zoology, NHM [email protected] Overview • Classical consensus – strict, loose, majority-rule, Adams • Within-consensus generalisations – reduced methods • Generalisation to supertrees – classical and reduced • Some (simple) examples • Which properties generalise and the meaning of loose supertrees Input Trees

Classical Consensus Trees Unique Plenary

Conservative Liberal strict/loose majority-rule (insensitive to weight) (sensitive to weight) Problems with Classical Consensus Methods

d a b c d

r r

b c d b c

1.0995 bits 1.58 bits r r ab nests within abcd (ab)c and/or (ab)d Reduced Consensus

Within-consensus generalisations 1. Define (e.g.) STRICT* as the set of (informative) strict consensus trees for each of the sets of trees induced from then set of input trees 2. [Define STRICT as the intersection of all largest compatible subsets of STRICT* ] 3. The strict reduced consensus profile is the set of non- redundant trees in STRICT Non-redundant = not implied (and sustained) by any sets of more inclusive trees…… Rhynchosaurs

H. gordoni H. huxleyi H. gordoni H. gordoni S. fischeri H. huxleyi S. sanjaunensis H. huxleyi S. fischeri S. fischeri Supradapedon S. sanjaunensis Nova Scotia S. sanjaunensis 1 Supradapedon 3 Supradapedon Texas Nova Scotia Isalorhnchus 2 Nova Scotia Isalorhnchus Isalorhnchus Acrodentus R. spenceri R. spenceri R. spenceri R. brodei R. brodei R. brodei R. articeps R. articeps R. articeps Mesodapedon Stenaulorhynchus Mesodapedon Stenaulorhynchus Stenaulorhynchus Howesia Howesia Mesosuchus Howesia Mesosuchus Mesosuchus

H. gordoni H. gordoni H. gordoni H. huxleyi 5 H. huxleyi 6 H. huxleyi S. fischeri S. fischeri S. fischeri S. sanjaunensis S. sanjaunensis S. sanjaunensis 4 Nova Scotia R. spenceri R. spenceri R. spenceri R. brodei R. brodei R. brodei R. articeps R. articeps R. articeps Mesodapedon Stenaulorhynchus Mesodapedon Stenaulorhynchus Howesia Stenaulorhynchus Howesia Mesosuchus Howesia Mesosuchus Mesosuchus Bootstrapping with Majority-rule Reduced Consensus X A B C D E J I H G F X A B C D E F G H I J

A 1111100000 B 0111100000 C 0011100000 50.5 50.5 D 0001100000 50.5 A B C D E X J I H G F E 0000100000 50.5 50.5 F 0000010000 G 0000011000 H 0000011100 I 0000011110 A B C D E F G H I J

A B C D E J I H G F J 0000011111 X 1111111111 100 99 98 99 100 98 Non-plenary 100 100 Non-unique INPUT TREES Majority- D E A C D E B C D A B C rule 1 2 3 4

Goloboff and Pol (2002), A B C D E A B C D E Goloboff (2006)

Majority-rule supertrees desirable in principle 5 = 2 + 3 + 4 6 = 1 + 3 + 4

MRP is a (poor) surrogate B C D E A C D E A B

Fundamental problem in generalising frequency of occurrence of groups 7 = 1 + 2 + 3 8 = 1 + 2 + 4 INPUT TREES Generalising D E A C D E B C D A B C

from 1 2 3 4 Objective A B D E A C D E B C D E A B C D

Functions x y x x x y 1 2 y 3 y 4 MR tree is the

strict A B C D E A B C D E

consensus of 4x 1x 100 1.5 majority-rule - the trees 2x, 3x, 4y 1x supertree 100 3.5 minimising the 2y, 3y 66 2 PRUNING sum of the Symmetric A B C D E Differences to semi-strict supertree

the input trees Strict/Semi-Strict(Loose)

S • Asymmetric strict refinement distance ( DX,Y) of X s from Y - If X and Y are irrelevant DX,Y = 0, if s X|L(Y) ⇒ Y| L(X), DX,Y is the minimum number of branch additions needed to convert Y|L(X) into s X|L(Y), otherwise DX,Y = ∞. L • Asymmetric loose refinement distance ( DX,Y) of L X from Y - If X and Y are irrelevant DX,Y = 0, if L X|L(Y) ⇒ Y| L(X), DX,Y is the minimum number of branch additions needed to convert Y|L(X) into a L tree that implies X|L(Y), otherwise DX,Y = ∞. r r

c a c y a r

1 3 5 7 r r r r y a c y b r r

6 8 r r r r

r r r

1, 2 7, 8

r c g f h g x h x z b b a a w w

A B

i g d h y g h x d e z

a w C x D e E b f

d e f g h x y z i F Properties/Interpretation • Loose/semi-strict – All and only the full splits • displayed (implied) by some tree(s) and – uncontradicted by any tree – uncontradicted by any set of compatible trees – uncontradicted by any set of compatible full splits • implied by the intersection of all largest compatible sets of displayed full splits (quartets) that are uncontradicted by any tree • What is implied by a set of trees after all conflict has been removed • “of displaying x(yz) if it is found in some input tree or implied by some combination of input trees and no input tree or combination of input trees displays or implies y(xz) or z(xy)” Goloboff and Pol (2002) With thanks….. • BBSRC • James Cotton • Organisers