## EMBOSS: PHYLIP

# Original PHYLIP dnapenny documentation.

This is the original PHYLIP documentation and includes information on the old menu driven program which is now replaced with the EMBOSS acd file.

                                                                       version 3.5c

                          DNAPENNY - Branch and bound to find
                              all most parsimonious trees
                     for nucleic acid sequence parsimony criteria

    (c) Copyright  1986-1993  by  Joseph  Felsenstein  and  by  the  University  of
    Washington.  Written by Joseph Felsenstein.  Permission is granted to copy this
    document provided that no fee is charged for it and that this copyright  notice
    is not removed.

         DNAPENNY is a program that will find all of the  most  parsimonious  trees
    implied  by  your  data  when  the nucleic acid sequence parsimony criterion is
    employed.  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  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).

                                     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.

                                 Controlling Run Times

         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.

                                  Method and Options

         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:

         1.  Each site evolves independently.

         2.  Different lineages evolve independently.

         3.  The probability of a base substitution at a given site is  small  over
    the lengths of time involved in a branch of the phylogeny.

         4.  The expected amounts of change in different branches of the  phylogeny
    do not vary by so much that two changes in a high-rate branch are more probable
    than one change in a low-rate branch.

         5.  The expected amounts of change do not vary enough among sites that two
    changes in one site are more probable than one change in another.

         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.

         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).

         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:

    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.

         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 documentation file for  the
    sequence programs.

         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.

    -------------------------------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     ......