The Algorithm
The search strategy used by DNAPENNY starts by making a tree consisting of the first two species (the first three if the tree is to be unrooted). Then it tries to add the next species in all possible places (there are three of these). For each of the resulting trees it evaluates the number of base substitutions. It adds the next species to each of these, again in all possible spaces. If this process would continue it would simply generate all possible trees, of which there are a very large number even when the number of species is moderate (34,459,425 with 10 species). Actually it does not do this, because the trees are generated in a particular order and some of them are never generated.
This is because the order in which trees are generated is not quite as implied above, but is a "depth-first search". This means that first one adds the third species in the first possible place, then the fourth species in its first possible place, then the fifth and so on until the first possible tree has been produced. For each tree the number of steps is evaluated. Then one "backtracks" by trying the alternative placements of the last species. When these are exhausted one tries the next placement of the next-to-last species. The order of placement in a depth-first search is like this for a four-species case (parentheses enclose monophyletic groups):
Make tree of first two species: (A,B)
Add C in first place: ((A,B),C)
Add D in first place: (((A,D),B),C)
Add D in second place: ((A,(B,D)),C)
Add D in third place: (((A,B),D),C)
Add D in fourth place: ((A,B),(C,D))
Add D in fifth place: (((A,B),C),D)
Add C in second place: ((A,C),B)
Add D in first place: (((A,D),C),B)
Add D in second place: ((A,(C,D)),B)
Add D in third place: (((A,C),D),B)
Add D in fourth place: ((A,C),(B,D))
Add D in fifth place: (((A,C),B),D)
Add C in third place: (A,(B,C))
Add D in first place: ((A,D),(B,C))
Add D in second place: (A,((B,D),C))
Add D in third place: (A,(B,(C,D)))
Add D in fourth place: (A,((B,C),D))
Add D in fifth place: ((A,(B,C)),D)
Among these fifteen trees you will find all of the four-species rooted trees,
each exactly once (the parentheses each enclose a monophyletic group). As
displayed above, the backtracking depth-first search algorithm is just another
way of producing all possible trees one at a time. The branch and bound
algorithm consists of this with one change. As each tree is constructed,
including the partial trees such as (A,(B,C)), its number of steps is
evaluated. In addition a prediction is made as to how many steps will be
added, at a minimum, as further species are added.
This is done by counting how many sites which are invariant in the data up the most recent species added will ultimately show variation when further species are added. Thus if 20 sites vary among species A, B, and C and their root, and if tree ((A,C),B) requires 24 steps, then if there are 8 more sites which will be seen to vary when species D is added, we can immediately say that no matter how we add D, the resulting tree can have no less than 24 + 8 = 32 steps. The point of all this is that if a previously-found tree such as ((A,B),(C,D)) required only 30 steps, then we know that there is no point in even trying to add D to ((A,C),B). We have computed the bound that enables us to cut off a whole line of inquiry (in this case five trees) and avoid going down that particular branch any farther.
The branch-and-bound algorithm thus allows us to find all most parsimonious trees without generating all possible trees. How much of a saving this is depends strongly on the data. For very clean (nearly "Hennigian") data, it saves much time, but on very messy data it will still take a very long time.
The algorithm in the program differs from the one outlined here in some essential details: it investigates possibilities in the order of their apparent promise. This applies to the order of addition of species, and to the places where they are added to the tree. After the first two-species tree is constructed, the program tries adding each of the remaining species in turn, each in the best possible place it can find. Whichever of those species adds (at a minimum) the most additional steps is taken to be the one to be added next to the tree. When it is added, it is added in turn to places which cause the fewest additional steps to be added. This sounds a bit complex, but it is done with the intention of eliminating regions of the search of all possible trees as soon as possible, and lowering the bound on tree length as quickly as possible. This process of evaluating which species to add in which order goes on the first time the search makes a tree; thereafter it uses that order.
The program keeps a list of all the most parsimonious trees found so far. Whenever it finds one that has fewer losses than these, it clears out the list and restarts it with that tree. In the process the bound tightens and fewer possibilities need be investigated. At the end the list contains all the shortest trees. These are then printed out. It should be mentioned that the program CLIQUE for finding all largest cliques also works by branch-and-bound. Both problems are NP-complete but for some reason CLIQUE runs far faster. Although their worst-case behavior is bad for both programs, those worst cases occur far more frequently in parsimony problems than in compatibility problems.
Among the quantities available to be set from the menu of DNAPENNY, two
(howoften and howmany) are of particular importance. As DNAPENNY goes along it
will keep count of how many trees it has examined. Suppose that howoften is
100 and howmany is 1000, the default settings. Every time 100 trees have been
examined, DNAPENNY will print out a line saying how many multiples of 100 trees
have now been examined, how many steps the most parsimonious tree found so far
has, how many trees of with that number of steps have been found, and a very
rough estimate of what fraction of all trees have been looked at so far.
When the number of these multiples printed out reaches the number howmany
(say 1000), the whole algorithm aborts and prints out that it has not found all
most parsimonious trees, but prints out what is has got so far anyway. These
trees need not be any of the most parsimonious trees: they are simply the most
parsimonious ones found so far. By setting the product (howoften X howmany)
large you can make the algorithm less likely to abort, but then you risk
getting bogged down in a gigantic computation. You should adjust these
constants so that the program cannot go beyond examining the number of trees
you are reasonably willing to pay for (or wait for). In their initial setting
the program will abort after looking at 100,000 trees. Obviously you may want
to adjust howoften in order to get more or fewer lines of intermediate notice
of how many trees have been looked at so far. Of course, in small cases you
may never even reach the first multiple of howoften, and nothing will be
printed out except some headings and then the final trees.
The indication of the approximate percentage of trees searched so far will
be helpful in judging how much farther you would have to go to get the full
search. Actually, since that fraction is the fraction of the set of all
possible trees searched or ruled out so far, and since the search becomes
progressively more efficient, the approximate fraction printed out will usually
be an underestimate of how far along the program is, sometimes a serious
underestimate.
A constant at the beginning of the program that affects the result is
"maxtrees", which controls the maximum number of trees that can be stored.
Thus if maxtrees is 25, and 32 most parsimonious trees are found, only the
first 25 of these are stored and printed out. If maxtrees is increased, the
program does not run any slower but requires a little more intermediate storage
space. I recommend that maxtrees be kept as large as you can, provided you are
willing to look at an output with that many trees on it! Initially, maxtrees
is set to 100 in the distribution copy.
The counting of the length of trees is done by an algorithm nearly
identical to the corresponding algorithms in DNAPARS, and thus the remainder of
this document will be nearly identical to the DNAPARS document.
This program carries out unrooted parsimony (analogous to Wagner trees)
(Eck and Dayhoff, 1966;
Kluge and Farris, 1969) on DNA sequences. The method
of
Fitch (1971) is used to count the number of changes of base needed on a
given tree. The assumptions of this method are exactly analogous to those of
DNAPARS:
That these are the assumptions of parsimony methods has been documented in a
series of papers of mine: (
1973a,
1978b,
1979,
1981b,
1983b,
1988b). For an
opposing view arguing that the parsimony methods make no substantive
assumptions such as these, see the papers by
Farris (1983) and
Sober (1983a, 1983b, 1988), but also read the exchange between
Felsenstein and Sober (1986).
Change from an occupied site to a deletion is counted as one change.
Reversion from a deletion to an occupied site is allowed and is also counted as
one change. Note that this in effect assumes that a deletion N bases long is N
separate events.
The input data is standard. The first line of the input file contains the
number of species and the number of sites. If the Weights option is being
used, there must also be a W in this first line to signal its presence. There
are only two options requiring information to be present in the input file, W
(Weights) and U (User tree). All options other than W (including U) are
invoked using the menu.
Next come the species data. Each sequence starts on a new line, has a
ten-character species name that must be blank-filled to be of that length,
followed immediately by the species data in the one-letter code. The sequences
must either be in the "interleaved" or "sequential" formats described in the
Molecular Sequence Programs document. The I option selects between them. The
sequences can have internal blanks in the sequence but there must be no extra
blanks at the end of the terminated line. Note that a blank is not a valid
symbol for a deletion.
The options are selected using an interactive menu. The menu looks like
this:
The options O, T, M, and 0 are the usual ones. They are described in the
main documentation file of this package. Option I is the same as in other
molecular sequence programs and is described in the
molecular sequence programs documentation.
The T (threshold) option allows a continuum of methods between parsimony
and compatibility. Thresholds less than or equal to 1.0 do not have any
meaning and should not be used: they will result in a tree dependent only on
the input order of species and not at all on the data!
The options H, F, and S are not found in the other molecular sequence
programs. H (How many) allows the user to set the quantity howmany, which we
have already seen controls number of times that the program will report on its
progress. F allows the user to set the quantity howoften, which sets how often
it will report -- after scanning how many trees.
The S (Simple) option alters a step in DNAPENNY which reconsiders the
order in which species are added to the tree. Normally the decision as to what
species to add to the tree next is made as the first tree is being constructed;
that ordering of species is not altered subsequently. The S option causes it
to be continually reconsidered. This will probably result in a substantial
increase in run time, but on some data sets of intermediate messiness it may
help. It is included in case it might prove of use on some data sets.
Output is standard: if option 1 is toggled on, the data is printed out,
with the convention that "." means "the same as in the first species". Then
comes a list of equally parsimonious trees, and (if option 2 is toggled on) a
table of the number of changes of state required in each character. If option
5 is toggled on, a table is printed out after each tree, showing for each
branch whether there are known to be changes in the branch, and what the states
are inferred to have been at the top end of the branch. If the inferred state
is a "?" or one of the IUB ambiguity symbols, there will be multiple equally-
parsimonious assignments of states; the user must work these out for themselves
by hand. A "?" in the reconstructed states means that in addition to one or
more bases, a deletion may or may not be present. If option 6 is left in its
default state the trees found will be written to a tree file, so that they are
available to be used in other programs.
Controlling Run Times
Method and Options
Penny algorithm for DNA, version 3.5c
branch-and-bound to find all most parsimonious trees
Settings for this run:
H How many groups of 100 trees: 1000
F How often to report, in trees: 100
S Branch and bound is simple? Yes
O Outgroup root? No, use as outgroup species 1
T Use Threshold parsimony? No, use ordinary parsimony
M Analyze multiple data sets? No
I Input sequences interleaved? Yes
0 Terminal type (IBM PC, VT52, ANSI)? ANSI
1 Print out the data at start of run No
2 Print indications of progress of run Yes
3 Print out tree Yes
4 Print out steps in each site No
5 Print sequences at all nodes of tree No
6 Write out trees onto tree file? Yes
Are these settings correct? (type Y or the letter for one to change)
The user either types "Y" (followed, of course, by a carriage-return) if the
settings shown are to be accepted, or the letter or digit corresponding to an
option that is to be changed.
TEST DATA SET
8 6
Alpha1 AAGAAG
Alpha2 AAGAAG
Beta1 AAGGGG
Beta2 AAGGGG
Gamma1 AGGAAG
Gamma2 AGGAAG
Delta GGAGGA
Epsilon GGAAAG
CONTENTS OF OUTPUT FILE (if all numerical options are on)
Penny algorithm for DNA, version 3.5c
branch-and-bound to find all most parsimonious trees
Name Sequences
---- ---------
Alpha1 AAGAAG
Alpha2 ......
Beta1 ...GG.
Beta2 ...GG.
Gamma1 .G....
Gamma2 .G....
Delta GGAGGA
Epsilon GGA...
requires a total of 8.000
9 trees in all found
+--------------------Alpha1
!
! +--Delta
! +--3
! +--7 +--Epsilon
--1 ! !
! +-----6 +-----Gamma2
! ! !
! +--4 +--------Gamma1
! ! !
! ! ! +--Beta2
+--2 +-----------5
! +--Beta1
!
+-----------------Alpha2
remember: this is an unrooted tree!
steps in each site:
0 1 2 3 4 5 6 7 8 9
*-----------------------------------------
0! 1 1 1 2 2 1
From To Any Steps? State at upper node
( . means same as in the node below it on tree)
1 AAGAAG
1 Alpha1 no ......
1 2 no ......
2 4 no ......
4 6 yes .G....
6 7 no ......
7 3 yes G.A...
3 Delta yes ...GGA
3 Epsilon no ......
7 Gamma2 no ......
6 Gamma1 no ......
4 5 yes ...GG.
5 Beta2 no ......
5 Beta1 no ......
2 Alpha2 no ......
+--------------------Alpha1
!
! +--Delta
! +-----3
! ! +--Epsilon
--1 +-----6
! ! ! +--Gamma2
! ! +-----7
! +--4 +--Gamma1
! ! !
! ! ! +--Beta2
+--2 +-----------5
! +--Beta1
!
+-----------------Alpha2
remember: this is an unrooted tree!
steps in each site:
0 1 2 3 4 5 6 7 8 9
*-----------------------------------------
0! 1 1 1 2 2 1
From To Any Steps? State at upper node
( . means same as in the node below it on tree)
1 AAGAAG
1 Alpha1 no ......
1 2 no ......
2 4 no ......
4 6 yes .G....
6 3 yes G.A...
3 Delta yes ...GGA
3 Epsilon no ......
6 7 no ......
7 Gamma2 no ......
7 Gamma1 no ......
4 5 yes ...GG.
5 Beta2 no ......
5 Beta1 no ......
2 Alpha2 no ......
+--------------------Alpha1
!
! +--Delta
! +--3
! +--6 +--Epsilon
--1 ! !
! +-----7 +-----Gamma1
! ! !
! +--4 +--------Gamma2
! ! !
! ! ! +--Beta2
+--2 +-----------5
! +--Beta1
!
+-----------------Alpha2
remember: this is an unrooted tree!
steps in each site:
0 1 2 3 4 5 6 7 8 9
*-----------------------------------------
0! 1 1 1 2 2 1
From To Any Steps? State at upper node
( . means same as in the node below it on tree)
1 AAGAAG
1 Alpha1 no ......
1 2 no ......
2 4 no ......
4 7 yes .G....
7 6 no ......
6 3 yes G.A...
3 Delta yes ...GGA
3 Epsilon no ......
6 Gamma1 no ......
7 Gamma2 no ......
4 5 yes ...GG.
5 Beta2 no ......
5 Beta1 no ......
2 Alpha2 no ......
+--------------------Alpha1
!
! +--Delta
! +--3
--1 +--7 +--Epsilon
! ! !
! +--------6 +-----Gamma2
! ! !
! ! +--------Gamma1
+--2
! +--Beta2
! +--5
+-----------4 +--Beta1
!
+-----Alpha2
remember: this is an unrooted tree!
steps in each site:
0 1 2 3 4 5 6 7 8 9
*-----------------------------------------
0! 1 1 1 2 2 1
From To Any Steps? State at upper node
( . means same as in the node below it on tree)
1 AAGAAG
1 Alpha1 no ......
1 2 no ......
2 6 yes .G....
6 7 no ......
7 3 yes G.A...
3 Delta yes ...GGA
3 Epsilon no ......
7 Gamma2 no ......
6 Gamma1 no ......
2 4 no ......
4 5 yes ...GG.
5 Beta2 no ......
5 Beta1 no ......
4 Alpha2 no ......
+--------------------Alpha1
!
! +--Delta
! +-----3
--1 ! +--Epsilon
! +--------6
! ! ! +--Gamma2
! ! +-----7
+--2 +--Gamma1
!
! +--Beta2
! +--5
+-----------4 +--Beta1
!
+-----Alpha2
remember: this is an unrooted tree!
steps in each site:
0 1 2 3 4 5 6 7 8 9
*-----------------------------------------
0! 1 1 1 2 2 1
From To Any Steps? State at upper node
( . means same as in the node below it on tree)
1 AAGAAG
1 Alpha1 no ......
1 2 no ......
2 6 yes .G....
6 3 yes G.A...
3 Delta yes ...GGA
3 Epsilon no ......
6 7 no ......
7 Gamma2 no ......
7 Gamma1 no ......
2 4 no ......
4 5 yes ...GG.
5 Beta2 no ......
5 Beta1 no ......
4 Alpha2 no ......
+--------------------Alpha1
!
! +--Delta
! +--3
--1 +--6 +--Epsilon
! ! !
! +--------7 +-----Gamma1
! ! !
! ! +--------Gamma2
+--2
! +--Beta2
! +--5
+-----------4 +--Beta1
!
+-----Alpha2
remember: this is an unrooted tree!
steps in each site:
0 1 2 3 4 5 6 7 8 9
*-----------------------------------------
0! 1 1 1 2 2 1
From To Any Steps? State at upper node
( . means same as in the node below it on tree)
1 AAGAAG
1 Alpha1 no ......
1 2 no ......
2 7 yes .G....
7 6 no ......
6 3 yes G.A...
3 Delta yes ...GGA
3 Epsilon no ......
6 Gamma1 no ......
7 Gamma2 no ......
2 4 no ......
4 5 yes ...GG.
5 Beta2 no ......
5 Beta1 no ......
4 Alpha2 no ......
+--------------------Alpha1
!
! +--Delta
! +--3
! +--7 +--Epsilon
--1 ! !
! +--6 +-----Gamma2
! ! !
! +-----2 +--------Gamma1
! ! !
+--4 +-----------Alpha2
!
! +--Beta2
+--------------5
+--Beta1
remember: this is an unrooted tree!
steps in each site:
0 1 2 3 4 5 6 7 8 9
*-----------------------------------------
0! 1 1 1 2 2 1
From To Any Steps? State at upper node
( . means same as in the node below it on tree)
1 AAGAAG
1 Alpha1 no ......
1 4 no ......
4 2 no ......
2 6 yes .G....
6 7 no ......
7 3 yes G.A...
3 Delta yes ...GGA
3 Epsilon no ......
7 Gamma2 no ......
6 Gamma1 no ......
2 Alpha2 no ......
4 5 yes ...GG.
5 Beta2 no ......
5 Beta1 no ......
+--------------------Alpha1
!
! +--Delta
! +-----3
! ! +--Epsilon
--1 +--6
! ! ! +--Gamma2
! +-----2 +-----7
! ! ! +--Gamma1
! ! !
+--4 +-----------Alpha2
!
! +--Beta2
+--------------5
+--Beta1
remember: this is an unrooted tree!
steps in each site:
0 1 2 3 4 5 6 7 8 9
*-----------------------------------------
0! 1 1 1 2 2 1
From To Any Steps? State at upper node
( . means same as in the node below it on tree)
1 AAGAAG
1 Alpha1 no ......
1 4 no ......
4 2 no ......
2 6 yes .G....
6 3 yes G.A...
3 Delta yes ...GGA
3 Epsilon no ......
6 7 no ......
7 Gamma2 no ......
7 Gamma1 no ......
2 Alpha2 no ......
4 5 yes ...GG.
5 Beta2 no ......
5 Beta1 no ......
+--------------------Alpha1
!
! +--Delta
! +--3
! +--6 +--Epsilon
--1 ! !
! +--7 +-----Gamma1
! ! !
! +-----2 +--------Gamma2
! ! !
+--4 +-----------Alpha2
!
! +--Beta2
+--------------5
+--Beta1
remember: this is an unrooted tree!
steps in each site:
0 1 2 3 4 5 6 7 8 9
*-----------------------------------------
0! 1 1 1 2 2 1
From To Any Steps? State at upper node
( . means same as in the node below it on tree)
1 AAGAAG
1 Alpha1 no ......
1 4 no ......
4 2 no ......
2 7 yes .G....
7 6 no ......
6 3 yes G.A...
3 Delta yes ...GGA
3 Epsilon no ......
6 Gamma1 no ......
7 Gamma2 no ......
2 Alpha2 no ......
4 5 yes ...GG.
5 Beta2 no ......
5 Beta1 no ......
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Maintained 15 Jul 1996 -- by Martin Hilbers(e-mail:M.P.Hilbers@dl.ac.uk)