- DESCRIPTION
- The Algorithm
- Controlling Run Times
- Methods and Options
- TEST DATA SET
- TEST SET OUTPUT (with all numerical options turned on)

There is, however, a price to be paid for the certainty that one has found all members of the set of most parsimonious trees. The problem of finding these has been shown (Graham and Foulds, 1982; Day, 1983) to be NP-complete, which is equivalent to saying that there is no fast algorithm that is guaranteed to solve the problem in all cases (for a discussion of NP- completeness, see the Scientific American article by Lewis and Papadimitriou, 1978). The result is that this program, despite its algorithmic sophistication, is VERY SLOW.

The program should be slower than the other tree-building programs in the package, but useable up to about ten species. Above this it will bog down rapidly, but exactly when depends on the data and on how much computer time you have (it may be more effective in the hands of someone who can let a microcomputer grind all night than for someone who has the "benefit" of paying for time on the campus mainframe computer). IT IS VERY IMPORTANT FOR YOU TO GET A FEEL FOR HOW LONG THE PROGRAM WILL TAKE ON YOUR DATA. This can be done by running it on subsets of the species, increasing the number of species in the run until you either are able to treat the full data set or know that the program will take unacceptably long on it. (Making a plot of the logarithm of run time against species number may help to project run times).

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 bifurcating 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 losses (or retentions of polymorphism) is evaluated.

The point of this is that if a previously-found tree such as ((A,B),(C,D)) required fewer losses, 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.

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 the list 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.

The counting of the length of trees is done by an algorithm nearly identical to the corresponding algorithms in DOLLOP, and thus the remainder of this document will be nearly identical to the DOLLOP document. The Dollo parsimony method was first suggested in print in verbal form by Le Quesne (1974) and was first well-specified by Farris (1977). The method is named after Louis Dollo since he was one of the first to assert that in evolution it is harder to gain a complex feature than to lose it. The algorithm explains

the presence of the state 1 by allowing up to one forward change 0-->1 and as many reversions 1-->0 as are necessary to explain the pattern of states seen. The program attempts to minimize the number of 1-->0 reversions necessary.

The assumptions of this method are in effect:

- We know which state is the ancestral one (state 0).
- The characters are evolving independently.
- Different lineages evolve independently.
- The probability of a forward change (0-->1) is small over the evolutionary times involved.
- The probability of a reversion (1-->0) is also small, but still far larger than the probability of a forward change, so that many reversions are easier to envisage than even one extra forward change.
- Retention of polymorphism for both states (0 and 1) is highly improbable.
- The lengths of the segments of the true tree are not so unequal that two changes in a long segment are as probable as one in a short segment.

One problem can arise when using additive binary recoding to represent a multistate character as a series of two-state characters. Unlike the Camin- Sokal, Wagner, and Polymorphism methods, the Dollo method can reconstruct ancestral states which do not exist. An example is given in my 1979 paper. It will be necessary to check the output to make sure that this has not occurred.

The polymorphism parsimony method was first used by me, and the results published (without a clear specification of the method) by Inger (1967). The method was published by Farris (1978a) and by me ( 1979). The method assumes that we can explain the pattern of states by no more than one origination (0-- >1) of state 1, followed by retention of polymorphism along as many segments of the tree as are necessary, followed by loss of state 0 or of state 1 where necessary. The program tries to minimize the total number of polymorphic characters, where each polymorphism is counted once for each segment of the tree in which it is retained.

The assumptions of the polymorphism parsimony method are in effect:

- The ancestral state (state 0) is known in each character.
- The characters are evolving independently of each other.
- Different lineages are evolving independently.
- Forward change (0-->1) is highly improbable over the length of time involved in the evolution of the group.
- Retention of polymorphism is also improbable, but far more probable that forward change, so that we can more easily envisage much polymorhism than even one additional forward change.
- Once state 1 is reached, reoccurrence of state 0 is very improbable, much less probable than multiple retentions of polymorphism.
- The lengths of segments in the true tree are not so unequal that we can more easily envisage retention events occurring in both of two long segments than one retention in a short segment.

The input format is the standard one, with "?", "P", "B" states allowed. Most of the options are selected using a menu:

Penny algorithm for Dollo or polymorphism parsimony, version 3.5c branch-and-bound to find all most parsimonious trees Settings for this run: P Parsimony method? Dollo H How many groups of 100 trees: 1000 F How often to report, in trees: 100 S Branch and bound is simple? Yes T Use Threshold parsimony? No, use ordinary parsimony A Use ancestral states in input file? No M Analyze multiple data sets? No 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 character No 5 Print states 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 P option toggles between the Polymorphism parsimony method and the default Dollo parsimony method.

The options T, A, and M are the usual Threshold, Ancestral States, and Multiple Data Sets options. They are described in the Main documentation file and in the Discrete Characters Programs documentation file.

Options F and H reset the variables howoften (F) and howmany (H). The user is prompted for the new values. By setting these larger the program will report its progress less often (howoften) and will run longer (howmany x often). These values default to 100 and 1000 which guarantees a search of 100,000 trees, but these can be changed. Note that option F in this program is not the Factors option available in some of the other programs in this section of the package.

The A (Ancestral states) option works in the usual way, described in the Discrete Characters Programs documentation file. It also requires that an A be present on the first line of the input file. The use of the A option allows implementation of the unordered Dollo parsimony and unordered polymorphism parsimony methods which I have described elsewhere ( 1984b). When the A option is used the ancestor is not to be counted as one of the species. The O (outgroup) option is not available since the tree produced is already rooted.

Setting T at or below 1.0 but above 0 causes the criterion to become compatibility rather than polymorphism parsimony, although there is no advantage to using this program instead of PENNY to do a compatibility method. Setting the threshold value higher brings about an intermediate between the Dollo or polymorphism parsimony methods and the compatibility method, so that there is some rationale for doing that. Since the Dollo and polymorphism methods produces a rooted tree, the user-defined trees required by the U option have two-way forks at each level.

Using a threshold value of 1.0 or lower, but above 0, one can obtain a rooted (or, if the A option is used with ancestral states of "?", unrooted) compatibility criterion, but there is no particular advantage to using this program for that instead of MIX. Higher threshold values are of course meaningful and provide intermediates between Dollo and compatibility methods.

The S (Simple) option alters a step in DOLPENNY 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 constructucted; that ordering of species is not altered subsequently. The R 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.

In the input file the W (Weights) option is available, as usual. It and the A (Ancestral states) option also requires the option to be declared on the first line of the input file, and other information to be present in the input file. If the Ancestral States option is used the A option must also be chosen from the menu. The F (Factors) option is not available in this program, as it would have no effect on the result even if that information were provided in the input file.

The output format is also standard. It includes a rooted tree and, if the user selects option 4, a table of the numbers of reversions or retentions of polymorphism necessary in each character. If any of the ancestral states has been specified to be unknown, a table of reconstructed ancestral states is also provided. When reconstructing the placement of forward changes and reversions under the Dollo method, keep in mind that each polymorphic state in the input data will require one "last minute" reversion. This is included in the tabulated counts. Thus if we have both states 0 and 1 at a tip of the tree the program will assume that the lineage had state 1 up to the last minute, and then state 0 arose in that population by reversion, without loss of state 1.

A table is available to be 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 "?" there will be multiple equally-parsimonious assignments of states; the user must work these out for themselves by hand.

If the A option is used, then the program will infer, for any character whose ancestral state is unknown ("?") whether the ancestral state 0 or 1 will give the best tree. If these are tied, then it may not be possible for the program to infer the state in the internal nodes, and these will all be printed as ".". If this has happened and you want to know more about the states at the internal nodes, you will find helpful to use DOLMOVE to display the tree and examine its interior states, as the algorithm in DOLMOVE shows all that can be known in this case about the interior states, including where there is and is not amibiguity. The algorithm in DOLPENNY gives up more easily on displaying these states.

At the beginning of the program are a series of constants, which can be changed to help adapt the program to different computer systems. Two are the initial values of howmany and howoften, constants "often" and "many". Constant "maxtrees" is the maximum number of tied trees that will be stored and "nmlngth" is the length of the species names.

7 6 Alpha1 110110 Alpha2 110110 Beta1 110000 Beta2 110000 Gamma1 100110 Delta 001001 Epsilon 001110

Penny algorithm for Dollo or polymorphism parsimony, version 3.5c branch-and-bound to find all most parsimonious trees Dollo parsimony method requires a total of 3.000 3 trees in all found +-----------------Delta ! --2 +--------------Epsilon ! ! +--3 +-----------Gamma1 ! ! +--6 +--------Alpha2 ! ! +--1 +--Beta2 ! +--5 +--4 +--Beta1 ! +-----Alpha1 reversions in each character: 0 1 2 3 4 5 6 7 8 9 *----------------------------------------- 0! 0 0 1 1 1 0 From To Any Steps? State at upper node ( . means same as in the node below it on tree) root 2 yes ..1.. . 2 Delta yes ..... 1 2 3 yes ...11 . 3 Epsilon no ..... . 3 6 yes 1.0.. . 6 Gamma1 no ..... . 6 1 yes .1... . 1 Alpha2 no ..... . 1 4 no ..... . 4 5 yes ...00 . 5 Beta2 no ..... . 5 Beta1 no ..... . 4 Alpha1 no ..... . +-----------------Delta ! --2 +--------------Epsilon ! ! +--3 +-----------Gamma1 ! ! +--6 +--Beta2 ! +-----5 ! ! +--Beta1 +--4 ! +--Alpha2 +-----1 +--Alpha1 reversions in each character: 0 1 2 3 4 5 6 7 8 9 *----------------------------------------- 0! 0 0 1 1 1 0 From To Any Steps? State at upper node ( . means same as in the node below it on tree) root 2 yes ..1.. . 2 Delta yes ..... 1 2 3 yes ...11 . 3 Epsilon no ..... . 3 6 yes 1.0.. . 6 Gamma1 no ..... . 6 4 yes .1... . 4 5 yes ...00 . 5 Beta2 no ..... . 5 Beta1 no ..... . 4 1 no ..... . 1 Alpha2 no ..... . 1 Alpha1 no ..... . +-----------------Delta ! --2 +--------------Epsilon ! ! +--3 +-----------Gamma1 ! ! ! ! +--Beta2 +--6 +--5 ! +--4 +--Beta1 ! ! ! +--1 +-----Alpha2 ! +--------Alpha1 reversions in each character: 0 1 2 3 4 5 6 7 8 9 *----------------------------------------- 0! 0 0 1 1 1 0 From To Any Steps? State at upper node ( . means same as in the node below it on tree) root 2 yes ..1.. . 2 Delta yes ..... 1 2 3 yes ...11 . 3 Epsilon no ..... . 3 6 yes 1.0.. . 6 Gamma1 no ..... . 6 1 yes .1... . 1 4 no ..... . 4 5 yes ...00 . 5 Beta2 no ..... . 5 Beta1 no ..... . 4 Alpha2 no ..... . 1 Alpha1 no ..... .

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