version 3.696

# Penny - Branch and bound to find
all most parsimonious trees

© Copyright 1986-2014 by Joseph Felsenstein. All rights reserved. License terms here.

Penny is a program that will find all of the most parsimonious trees implied by your data. It does so not by examining all possible trees, but by using the more sophisticated "branch and bound" algorithm, a standard computer science search strategy first applied to phylogenetic inference by Hendy and Penny (1982). (J. S. Farris [personal communication, 1975] had also suggested that this strategy, which is well-known in computer science, might be applied to phylogenies, but he did not publish this suggestion).

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 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).

## The Algorithm

The search strategy used by Penny 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 steps. 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.

Actually 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. Its 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 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 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 binary characters which are invariant in the data up the species most recently added will ultimately show variation when further species are added. Thus if 20 characters 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 characters 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.

The program keeps a list of all the most parsimonious trees found so far. Whenever it finds one that has fewer steps 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.

## Controlling Run Times

Among the quantities available to be set at the beginning of a run of Penny, two (howoften and howmany) are of particular importance. As Penny 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, Penny 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 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 times 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 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.

## Methods and Options

The counting of the length of trees is done by an algorithm nearly identical to the corresponding algorithms in Mix, and thus the remainder of this document will be nearly identical to the Mix document. Mix is a general parsimony program which carries out the Wagner and Camin-Sokal parsimony methods in mixture, where each character can have its method specified. The program defaults to carrying out Wagner parsimony.

The Camin-Sokal parsimony method explains the data by assuming that changes 0 --> 1 are allowed but not changes 1 --> 0. Wagner parsimony allows both kinds of changes. (This under the assumption that 0 is the ancestral state, though the program allows reassignment of the ancestral state, in which case we must reverse the state numbers 0 and 1 throughout this discussion). The criterion is to find the tree which requires the minimum number of changes. The Camin-Sokal method is due to Camin and Sokal (1965) and the Wagner method to Eck and Dayhoff (1966) and to Kluge and Farris (1969).

Here are the assumptions of these two methods:

1. Ancestral states are known (Camin-Sokal) or unknown (Wagner).
2. Different characters evolve independently.
3. Different lineages evolve independently.
4. Changes 0 --> 1 are much more probable than changes 1 --> 0 (Camin-Sokal) or equally probable (Wagner).
5. Both of these kinds of changes are a priori improbable over the evolutionary time spans involved in the differentiation of the group in question.
6. Other kinds of evolutionary event such as retention of polymorphism are far less probable than 0 --> 1 changes.
7. Rates of evolution in different lineages are sufficiently low that two changes in a long segment of the tree are far less probable than one change in a short segment.

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), but also read the exchange between Felsenstein and Sober (1986).

The input for Penny is the standard input for discrete characters programs, described above in the documentation file for the discrete-characters programs. States "?", "P", and "B" are allowed.

The options are selected using a menu:

    Penny algorithm, version 3.696
     branch-and-bound to find all most parsimonious trees

    Settings for this run:
      X                     Use Mixed method?  No
      P                     Parsimony method?  Wagner
      F        How often to report, in trees:  100
      H        How many groups of  100 trees:  1000
      O                        Outgroup root?  No, use as outgroup species  1
      S           Branch and bound is simple?  Yes
      T              Use Threshold parsimony?  No, use ordinary parsimony
      A   Use ancestral states in input file?  No
      W                       Sites weighted?  No
      M           Analyze multiple data sets?  No
      0   Terminal type (IBM PC, ANSI, none)?  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 options X, O, T, A, and M are the usual miXed Methods, Outgroup, Threshold, Ancestral States, and Multiple Data Sets options. They are described in the Main documentation file and in the Discrete Characters Programs documentation file. The O option is only acted upon if the final tree is unrooted.

The option P toggles between the Camin-Sokal parsimony criterion and the Wagner parsimony criterion. 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 times howoften). 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. If the A option is not used, then the program will assume 0 as the ancestral state for those characters following the Camin-Sokal method, and will assume that the ancestral state is unknown for those characters following Wagner parsimony. If any characters have unknown ancestral states, and if the resulting tree is rooted (even by outgroup), a table will be printed out showing the best guesses of which are the ancestral states in each character.

The S (Simple) option alters a step in Penny 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. The Simple option, in which the ordering is kept the same after being established by trying alternatives during the construction of the first tree, is the default. Continual reconsideration can be selected as an alternative.

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 final output is standard: a set of trees, which will be printed as rooted or unrooted depending on which is appropriate, and if the user elects to see them, tables of the number of changes of state required in each character. If the Wagner option is in force for a character, it may not be possible to unambiguously locate the places on the tree where the changes occur, as there may be multiple possibilities. 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 Camin-Sokal parsimony method (option C or S) is invoked and the A option is also used, then the program will infer, for any character whose ancestral state is unknown ("?") whether the ancestral state 0 or 1 will give the fewest state changes. 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 Move to display the tree and examine its interior states, as the algorithm in Move shows all that can be known in this case about the interior states, including where there is and is not amibiguity. The algorithm in Penny gives up more easily on displaying these states.

If the A option is not used, then the program will assume 0 as the ancestral state for those characters following the Camin-Sokal method, and will assume that the ancestral state is unknown for those characters following Wagner parsimony. If any characters have unknown ancestral states, and if the resulting tree is rooted (even by outgroup), a table will be printed out showing the best guesses of which are the ancestral states in each character. You will find it useful to understand the difference between the Camin-Sokal parsimony criterion with unknown ancestral state and the Wagner parsimony criterion.

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. If the program finds multiple trees tied for best, all of these are written out onto the output tree file. Each is followed by a numerical weight in square brackets (such as [0.25000]). This is needed when we use the trees to make a consensus tree of the results of bootstrapping or jackknifing, to avoid overrepresenting replicates that find many tied trees.

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.

### TEST DATA SET

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

### TEST SET OUTPUT (with all numerical options turned on)

    Penny algorithm, version 3.69
     branch-and-bound to find all most parsimonious trees

     7 species,   6 characters
    Wagner parsimony method

    Name         Characters
    ----         ----------

    Alpha1       11011 0
    Alpha2       11011 0
    Beta1        11000 0
    Beta2        11000 0
    Gamma1       10011 0
    Delta        00100 1
    Epsilon      00111 0

    requires a total of              8.000

        3 trees in all found

      +-----------------Alpha1
      !
      !        +--------Alpha2
    --1        !
      !  +-----4     +--Epsilon
      !  !     !  +--6
      !  !     +--5  +--Delta
      +--2        !
         !        +-----Gamma1
         !
         !           +--Beta2
         +-----------3
                     +--Beta1

      remember: this is an unrooted tree!

    steps in each character:
             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                11011 0
      1    Alpha1       no     ..... .
      1       2         no     ..... .
      2       4         no     ..... .
      4    Alpha2       no     ..... .
      4       5         yes    .0... .
      5       6         yes    0.1.. .
      6    Epsilon      no     ..... .
      6    Delta        yes    ...00 1
      5    Gamma1       no     ..... .
      2       3         yes    ...00 .
      3    Beta2        no     ..... .
      3    Beta1        no     ..... .

      +-----------------Alpha1
      !
    --1  +--------------Alpha2
      !  !
      !  !           +--Epsilon
      +--2        +--6
         !  +-----5  +--Delta
         !  !     !
         +--4     +-----Gamma1
            !
            !        +--Beta2
            +--------3
                     +--Beta1

      remember: this is an unrooted tree!

    steps in each character:
             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                11011 0
      1    Alpha1       no     ..... .
      1       2         no     ..... .
      2    Alpha2       no     ..... .
      2       4         no     ..... .
      4       5         yes    .0... .
      5       6         yes    0.1.. .
      6    Epsilon      no     ..... .
      6    Delta        yes    ...00 1
      5    Gamma1       no     ..... .
      4       3         yes    ...00 .
      3    Beta2        no     ..... .
      3    Beta1        no     ..... .

      +-----------------Alpha1
      !
      !           +-----Alpha2
    --1  +--------2
      !  !        !  +--Beta2
      !  !        +--3
      +--4           +--Beta1
         !
         !           +--Epsilon
         !        +--6
         +--------5  +--Delta
                  !
                  +-----Gamma1

      remember: this is an unrooted tree!

    steps in each character:
             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                11011 0
      1    Alpha1       no     ..... .
      1       4         no     ..... .
      4       2         no     ..... .
      2    Alpha2       no     ..... .
      2       3         yes    ...00 .
      3    Beta2        no     ..... .
      3    Beta1        no     ..... .
      4       5         yes    .0... .
      5       6         yes    0.1.. .
      6    Epsilon      no     ..... .
      6    Delta        yes    ...00 1
      5    Gamma1       no     ..... .