lecture 2. tree space and searching tree...

Lecture 2. Tree space and searching tree space

Joe Felsenstein

Department of Genome Sciences and Department of Biology

Lecture 2. Tree space and searching tree space – p.1/48

The same tree?

Hum

an

Chi

mp

Gor

illa

Ora

ng

Gib

bon

Mac

acqu

e

Col

obus

Hum

an

Chi

mp

Gor

illa

Ora

ng

Gib

bon

Mac

acqu

e

Col

obus


All possible treesa b

Forming all 4-species trees by adding the next species in all possibleplaces



a bca bc bc a




a bca bc bc a

a d c b

c bd a

c b

c

a d

a b d

a c b d

etc. etc.



The number of rooted bifurcating trees:

1 × 3 × 5 × 7 × . . . × (2n − 3)

= (2n − 3)!/(

(n − 2)! 2n−2)


which is:species number of trees

1 12 13 34 155 1056 9457 10,3958 135,1359 2,027,025

10 34,459,42511 654,729,07512 13,749,310,57513 316,234,143,22514 7,905,853,580,62515 213,458,046,676,87516 6,190,283,353,629,37517 191,898,783,962,510,62518 6,332,659,870,762,850,62519 221,643,095,476,699,771,87520 8,200,794,532,637,891,559,37530 4.9518 ×10

38

40 1.00985 ×1057

50 2.75292 ×1076


Mapping an unrooted tree into a rooted tree

12

3

4

5 6

7

8

1

6

47

5

2

8 3

12

3

4

5 6

7

8

1

6

47

5

2

8 3

... one with one fewer species.


For one tree topology

The space of trees varying all 2n − 3 branch lengths, each a nonegativenumber, defines an “orthant" (open corner) of a (2n − 3)-dimensional realspace:

A

B

C

D

E

F

v 1

v

vv

v

v

v 23

v 4

5

6

78

v 9

wall wall

floor v 9


Through the looking glass ...

A

B

C

D

E

F

v 1

v

vv

v

v

v 23

v 4

5

6

78

v 9



A

B

C

D

E

F

v 1

v

vv

v

v

v 23

v 4

5

6

78

v 9

A

B

C

E

F

v 1

v

vv

vv 2

3

D

v

v 4

56

78



A

B

C

D

E

F

v 1

v

vv

v

v

v 23

v 4

5

6

78

v 9

A

B

C

E

F

v 1

v

vv

vv 2

3

D

v

v 4

56

78

A

B

C

D

E

F

v 1

v

vv

v

v

v 23

v 4

56

78

v 9



A

B

C

D

E

F

v 1

v

vv

v

v

v 23

v 4

5

6

78

v 9

A

B

C

E

F

v 1

v

vv

vv 2

3

D

v

v 4

56

78

A

B

C

D

E

F

v 1

v

v

vv

v

v 23

v 4

5

6

7

8

v 9


Tree space

t1t2

t1t2

an example: three species with a clock

A B C

t 1

t 2

t 1

t 2

OK

not possible

trifurcation

etc.

when we consider all three possible topologies, the space looks like:


A global maximum is not easy to find

If start here



If start here



end up here

If start here



end up here but global maximum is here

If start here


Nearest-neighbor interchanges (NNIs)

U V

U V

U V

S T

S T

S T S T

U V

and reforming them in one of the two possible alternative ways:

is rearranged by dissolving the connections to an interior branch

A subtree

(The triangles are subtrees)Lecture 2. Tree space and searching tree space – p.31/48

all 15 trees, connected by NNIs

D B

C E

A

C E

D A

B D C

A E

B

A C

D E

B

E B

C D

A B C

D E

A

C B

D E

A

A B

D E

C

E B

D C

AB D

C E

A

B C

E D

A

A B

E C

D

D B

E C

AC D

B E

A

E C

B D

A

The graph of all 15 5-species unrooted trees, connected by NNIs


with parsimony scores

11

9

11

11

11

10

11

9

11

9

11

8

9

10

9

The same graph with parsimony scores (try a “greedy" search with NNI’s)


Subtree pruning and regrafting (SPR) rearrangement

A

BC

D

E

F

G

H

I

J

KA

BC

D

E

F

G

H

I

J

K

Break a branch, remove a subtree

Add it in, attaching it to one (*)

A

C

D

E

G

K

*

A

C

D

G

E

K

B

F

H

I

J

Here is the result:

of the other branches


Tree bisection and reconnection (TBR) rearrangement

A

BC

D

E

F

G

H

I

J

KA

BC

D

E

F

G

H

I

J

K

Here is the result:

Break a branch, separate the subtrees

Connect a branch of one

A

BC

D

E

F

G

H

I

J

KA

B

C

D

E

GF

HI

J

K

to a branch of the other


Greedy search by sequential addition

A

D

B

C

A

B

C

B

C

D

A

D

A

C

B

8 7 9

BA

D

C

E11

A

D 9

E

C

B

A

D E9

BC

A

9 C

B

E

D

D 9 C

BEA

Greedy search by addition of species in a fixed order (A, B, C, D, E) in thebest place each time. Lecture 2. Tree space and searching tree space – p.36/48

Goloboff’s time-saving trick

H−K

L

M−

R

S−U

A

V−Z

V−Z

A−G H−R

S−U

B−G

Goloboff’s economy in computing scores of rearranged treesOnce the “views” have been computed, they can be taken to

represent subtrees, without going inside those subtrees


Star decomposition

A

C

D

E F

B

E

C

D

A

B

F

B C

D

A

E F

E

C

D

A

B

F

B C

D

A

F

E

C

D

A

B

F

E

“Star decomposition" search for best tree can happen in multiple ways


Disk-covering

A

B

C D

EF

0.1

0.05

0.1 0.04 0.1

0.030.030.02

0.05

“Disk covering" – assembly of a tree from overlapping estimated subtrees


Shortest Hamiltonian path problem(a) (b)

(c) (d)


Search tree for this problem

etc. etc.

etc.etc.

add 1 add 2 add 3

add 2 add 3 add 4 add 5

add 3 add 5

add 8 add 10add 9

add 9

add 9add 3add 10

add 10 add 8

add 8add 3add 10

add 10 add 8

add 8add 3

add 9

etc. etc.

start

(1,2,3,4,5,6,7,8,9,10) (1,2,3,4,5,6,7,9,8,10) (1,2,3,4,5,6,7,10,8,9)

(1,2,3,4,5,6,7,8,10,9) (1,2,3,4,5,6,7,9,10,8) (1,2,3,4,5,6,7,10,9,8)

add 4

etc.

add 9


Search tree of trees

C

A

B

D C

A B

A

C

B

D

A

B

C

D

A

ED

B

C

E

DA

B

C

D

AE

B

C

D

AC

B

E

D

AB

C

E

E

AC

B

D

E

C A B

DC

AE

B

D

C

AD

B

E

C

AB

D

E

E

AB

C

D

E

BA

C

D

B

AE

C

D

B

AD

C

E

B

A C D

ELecture 2. Tree space and searching tree space – p.42/48

same, with parsimony scores in place of trees

8

11

11

9

3

9

7 8

9

9

9

10

10

1111

11

11

9

11


Polynomial time and exponential time

1 10 10010

0

101

102

103

104

105

106

Tim

e

Problem size

6n +4n−33

e0.5n

How does the time taken by an algorithm depend on the size of theproblem? If it is a polynomial (even one with big coefficients), with a bigenough case it is faster than one that depends on the size exponentially.


NP completeness and NP hardness

P

NP

does thispart exist?

NP Hard

is P = NP?

NP Complete

(This diagram is not quite correct – see the diagrams on the Wikipedia page for “NP-hard”).

P = problems that can be solved by a polynomial time algorithm

NP complete = problems for which a proposed solution can be checked in polynomial timebut for which it can be proven that if one of them is in P, all are.

NP hard = problems for which a solution can be checked in polynomial time, but might be notsolvable in polynomial time.


Some referencesFelsenstein, J. 1978. The number of evolutionary trees. Systematic Zoology27: 27-33.

(Correction, vol. 30, p. 122, 1981) [Review of counting tip-labelled trees, recursion forcounting multifurcating case]

Cavalli-Sforza, L. L. and A. W. F. Edwards. 1967. Phylogenetic analysis: models and estimationprocedures. American Journal of Human Genetics19: 233-257. also Evolution21: 550-570.[Includes counting and tree shapes]

Camin, J. H. and R. R. Sokal. 1965. A method for deducing branching sequences in phylogeny.Evolution19: 311-326. [Early parsimony paper includes rearrangement of trees]

Waterman, M. S. and T. F. Smith. 1978. On the similarity of dendrograms. Journal of Theoretical

Biology73: 789-800. [Defines NNIs. Uses them to get a distance between trees.]Maddison, D. R. 1991. The discovery and importance of multiple islands of most-parsimonious

trees. Systematic Zoology40: 315-328. [Discusses heuristic search strategy involving ties,multiple starts]

Farris, J. S. 1970. Methods for computing Wagner trees. Systematic Zoology19: 83-92. [Earlyparsimony algorithms paper is one of first to mention sequential addition strategy]


continuedSaitou, N., and M. Nei. 1987. The neighbor-joining method: a new method for reconstructing

phylogenetic trees. Molecular Biology and Evolution4: 406-425. [First mention ofstar-decomposition search for best trees, sort of]

Strimmer, K., and A. von Haeseler. 1996. Quartet puzzling: a quartet maximum likelihoodmethod for reconstructing tree topologies. Molecular Biology and Evolution13: 964-969.[Assembles trees out of quartets]

Huson, D., S. Nettles, L. Parida, T. Warnow, and S. Yooseph. 1998. The disk-covering method fortree reconstruction. pp. 62-75 in Proceedings of “Algorithms and Experiments” (ALEX98), Trento,

Italy, Feb. 9-11, 1998, ed. R. Battiti and A. A. Bertossi. [“Disk-covering method” for longstringy trees]

Swofford, D. L. and G. J. Olsen. 1990. Phylogeny reconstruction. Chapter 11, Pp. 411-501 inMolecular Systematics,ed. D. M. Hillis and C. Moritz. Sinauer Associates, Sunderland,Massachusetts. [Review that discusses strategies, names SPR and TBR rearrangementmethods]

Foulds, L. R. and R. L. Graham. 1982. The Steiner problem in phylogeny is NP-complete.Advances in Applied Mathematics3: 43-49. [Parsimony is NP-hard]

Graham, R. L. and L. R. Foulds. 1982. Unlikelihood that minimal phylogenies for a realisticbiological study can be constructed in reasonable computat ional time. Mathematical

Biosciences60: 133-142. [ ... and more]


continuedHendy, M. D. and D. Penny. 1982. Branch and bound algorithms to determine minimal

evolutionary trees. Mathematical Biosciences60: 133-142 [Introduced branch-and-bound forphylogenies]

Felsenstein, J. 2004. Inferring Phylogenies.Sinauer Associates, Sunderland, Massachusetts. [Forthis lecture the material is chapters 3, 4, and 5]

Semple, C. and M. Steel. 2003. Phylogenetics.Oxford University Press, Oxford. [Also coverssearch strategies]


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