Lecture 2. Tree space and searching tree space
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1 Lecture 2. Tree space and searching tree space Joe Felsenstein epartment of Genome Sciences and epartment of iology Lecture 2. Tree space and searching tree space p.1/48
2 Orang Gorilla himp Human Gibbon Macacque olobus Macacque Gibbon Gorilla himp Human Orang olobus The same tree? Lecture 2. Tree space and searching tree space p.2/48
3 ll possible trees a b Forming all 4-species trees by adding the next species in all possible places Lecture 2. Tree space and searching tree space p.3/48
4 ll possible trees a b a c b a c b c a b Forming all 4-species trees by adding the next species in all possible places Lecture 2. Tree space and searching tree space p.4/48
5 ll possible trees a b a c b a c b c a b etc. etc. a d c b a c b d d a c b a b c d a d c b Forming all 4-species trees by adding the next species in all possible places Lecture 2. Tree space and searching tree space p.5/48
6 The number of rooted bifurcating trees: (2n 3) = (2n 3)!/ ( (n 2)! 2 n 2) Lecture 2. Tree space and searching tree space p.6/48
7 which is: species number of trees , , ,027, ,459, ,729, ,749,310, ,234,143, ,905,853,580, ,458,046,676, ,190,283,353,629, ,898,783,962,510, ,332,659,870,762,850, ,643,095,476,699,771, ,200,794,532,637,891,559, Lecture 2. Tree space and searching tree space p.7/48
8 Mapping an unrooted tree into a rooted tree one with one fewer species. Lecture 2. Tree space and searching tree space p.8/48
9 For one tree topology The space of trees varying all 2n 3 branch lengths, each a nonegative number, defines an orthant" (open corner) of a (2n 3)-dimensional real space: v 1 F v 6 v 2 3 wall 8v 9 v v v 7 v 5 v 4 floor wall v 9 Lecture 2. Tree space and searching tree space p.9/48
10 Through the looking glass... v 1 F v 6 v v v 8 7 v 2 3 v 9 v 5 v 4 Lecture 2. Tree space and searching tree space p.10/48
11 Through the looking glass... v 1 F v 6 v 2 3 v v v 8 7 v 9 v 5 v 4 v 1 F v 6 v 2 3 v v v 8 7 v 5 v 4 Lecture 2. Tree space and searching tree space p.11/48
12 Through the looking glass... v 1 F v 6 v 2 3 v v v 8 7 v 9 v 5 v 4 v 1 F v 6 v 7 v 2 v3 v 8 v 4 v 9 v 5 v 1 F v 6 v 2 3 v v v 8 7 v 5 v 4 Lecture 2. Tree space and searching tree space p.12/48
13 Through the looking glass... v 1 F v 6 v 2 3 v v v 8 7 v 9 v 5 v 4 v 1 F v 6 v 2 3 v v v 8 7 v 5 v 4 v 1 v 7 v 9 v 2 v3 v8 v 5 F v 6 v 4 Lecture 2. Tree space and searching tree space p.13/48
14 Tree space an example: three species with a clock trifurcation t 1 t 2 t 1 not possible OK etc. t 2 when we consider all three possible topologies, the space looks like: t 1 t 2 Lecture 2. Tree space and searching tree space p.14/48
15 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.15/48
16 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.16/48
17 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.17/48
18 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.18/48
19 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.19/48
20 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.20/48
21 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.21/48
22 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.22/48
23 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.23/48
24 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.24/48
25 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.25/48
26 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.26/48
27 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.27/48
28 global maximum is not easy to find If start here Lecture 2. Tree space and searching tree space p.28/48
29 global maximum is not easy to find end up here If start here Lecture 2. Tree space and searching tree space p.29/48
30 global maximum is not easy to find end up here but global maximum is here If start here Lecture 2. Tree space and searching tree space p.30/48
31 Nearest-neighbor interchanges (NNIs) subtree S T U V is rearranged by dissolving the connections to an interior branch S T U V and reforming them in one of the two possible alternative ways: S T S T U V U V (The triangles are subtrees) Lecture 2. Tree space and searching tree space p.31/48
32 all 15 trees, connected by NNIs Lecture 2. Tree space and searching tree space p.32/48
33 with parsimony scores Lecture 2. Tree space and searching tree space p.33/48
34 Subtree pruning and regrafting (SPR) rearrangement F G H I J K F G H I J K reak a branch, remove a subtree dd it in, attaching it to one (*) G K * G K F H I J Here is the result: of the other branches Lecture 2. Tree space and searching tree space p.34/48
35 Tree bisection and reconnection (TR) rearrangement F G H I J K F G H I J K Here is the result: reak a branch, separate the subtrees onnect a branch of one F G H I J K G F H I J K to a branch of the other Lecture 2. Tree space and searching tree space p.35/48
36 Greedy search by sequential addition Greedy search by addition of species in a fixed order (,,,, ) in the best place each time. Lecture 2. Tree space and searching tree space p.36/48
37 Goloboff s time-saving trick G H K G H R L V Z S U V Z M R S U Goloboff s economy in computing scores of rearranged trees Once the views have been computed, they can be taken to represent subtrees, without going inside those subtrees Lecture 2. Tree space and searching tree space p.37/48
38 Star decomposition F F F F F F Star decomposition" search for best tree can happen in multiple ways Lecture 2. Tree space and searching tree space p.38/48
39 isk-covering F isk covering" assembly of a tree from overlapping estimated subtrees Lecture 2. Tree space and searching tree space p.39/48
40 Shortest Hamiltonian path problem (a) (b) (c) (d) Lecture 2. Tree space and searching tree space p.40/48
41 Search tree for this problem (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) (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) add add 310 add 9 add add 310 add 8 add add 39 add 8 add 9 add 10 add 8 add 10 add 8 add 9 add 8 add 10 add 9 etc. etc. add 3 add 4 add 5 etc. etc. etc. add 2 add 3 add 4 add 5 etc. etc. add 1 add 2 add 3 start Lecture 2. Tree space and searching tree space p.41/48
42 Search tree of trees Lecture 2. Tree space and searching tree space p.42/48
43 same, with parsimony scores in place of trees Lecture 2. Tree space and searching tree space p.43/48
44 Time Polynomial time and exponential time n +4n 3 e 0.5n Problem size How does the time taken by an algorithm depend on the size of the problem? If it is a polynomial (even one with big coefficients), with a big enough case it is faster than one that depends on the size exponentially. Lecture 2. Tree space and searching tree space p.44/48
45 NP completeness and NP hardness P NP does this part exist? is P = NP? NP Hard NP omplete (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 time but 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 not solvable in polynomial time. Lecture 2. Tree space and searching tree space p.45/48
46 Some references Felsenstein, J The number of evolutionary trees. Systematic Zoology 27: (orrection, vol. 30, p. 122, 1981) [Review of counting tip-labelled trees, recursion for counting multifurcating case] avalli-sforza, L. L. and. W. F. dwards Phylogenetic analysis: models and estimation procedures. merican Journal of Human Genetics 19: also volution 21: [Includes counting and tree shapes] amin, J. H. and R. R. Sokal method for deducing branching sequences in phylogeny. volution 19: [arly parsimony paper includes rearrangement of trees] Waterman, M. S. and T. F. Smith On the similarity of dendrograms. Journal of Theoretical iology 73: [efines NNIs. Uses them to get a distance between trees.] Maddison,. R The discovery and importance of multiple islands of most-parsimonious trees. Systematic Zoology 40: [iscusses heuristic search strategy involving ties, multiple starts] Farris, J. S Methods for computing Wagner trees. Systematic Zoology 19: [arly parsimony algorithms paper is one of first to mention sequential addition strategy] Lecture 2. Tree space and searching tree space p.46/48
47 continued Saitou, N., and M. Nei The neighbor-joining method: a new method for reconstructing phylogenetic trees. Molecular iology and volution 4: [First mention of star-decomposition search for best trees, sort of] Strimmer, K., and. von Haeseler Quartet puzzling: a quartet maximum likelihood method for reconstructing tree topologies. Molecular iology and volution 13: [ssembles trees out of quartets] Huson,., S. Nettles, L. Parida, T. Warnow, and S. Yooseph The disk-covering method for tree reconstruction. pp in Proceedings of lgorithms and xperiments (LX98), Trento, Italy, Feb. 9-11, 1998, ed. R. attiti and.. ertossi. [ isk-covering method for long stringy trees] Swofford,. L. and G. J. Olsen Phylogeny reconstruction. hapter 11, Pp in Molecular Systematics, ed.. M. Hillis and. Moritz. Sinauer ssociates, Sunderland, Massachusetts. [Review that discusses strategies, names SPR and TR rearrangement methods] Foulds, L. R. and R. L. Graham The Steiner problem in phylogeny is NP-complete. dvances in pplied Mathematics 3: [Parsimony is NP-hard] Graham, R. L. and L. R. Foulds Unlikelihood that minimal phylogenies for a realistic biological study can be constructed in reasonable computat ional time. Mathematical iosciences 60: [... and more] Lecture 2. Tree space and searching tree space p.47/48
48 continued Hendy, M.. and. Penny ranch and bound algorithms to determine minimal evolutionary trees. Mathematical iosciences 60: [Introduced branch-and-bound for phylogenies] Felsenstein, J Inferring Phylogenies. Sinauer ssociates, Sunderland, Massachusetts. [For this lecture the material is chapters 3, 4, and 5] Semple,. and M. Steel Phylogenetics. Oxford University Press, Oxford. [lso covers search strategies] Lecture 2. Tree space and searching tree space p.48/48
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