version 3.5c


         PENNY - Branch and bound to find all most parsimonious trees


(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.

     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  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 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 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 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.

     Most of the options are selected using a menu:












Penny algorithm, version 3.5c
 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
  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 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 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, and requires one (or more)
lines of auxiliary information  specifying  the  states  of  the  ancestor,  as
described  in those documentation files.  It also requires that an A be present
on the first line of the input file.  The ancestor is not to be counted as  one
of the species.  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
constructucted;  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.

     In the input file the W (Weights) option is available, as  usual.   The  A
(Ancestral states) and X (Mixed methods) also require information to be present


and options declared on the first line of the input  file.   If  the  Ancestral
States  option is invoked then you must also choose A from the menu.  Note that
the X option requires that the option M (Mixture) be declared and  the  mixture
information  provided  in the input file.  This is admittedly confusing.  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.

     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.

-------------------------------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 3.5c


 branch-and-bound to find all most parsimonious trees

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     ..... .