Evolution__3rd_Edition

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The number of possible trees can be
astronomically large
Algorithms are used to search a
subsample of trees
of events implied by them all. For five species, however, 15 trees are possible. The general
formula for the number of possible unrooted bifurcating trees for s species is:
Number of possible unrooted trees =
CHAPTER 15 / The Reconstruction of Phylogeny 453
s
∏
i=
3
( 2i−5) The Π term means “product”: we multiply (that is, take the product of) all the possible
terms in the parentheses. For three species, s = 3 and there is only one term to take the
product of (from i = 3 up to s, which is also 3); the parenthetic term for i = 3 is 6 − 5 = 1,
and the number of possible trees is therefore one. For s = 4, we have to multiply that 1
by the parenthetic term for i = 4, which is 3; 3 × 1 = 3, the number of unrooted trees for
four species. For s = 5, the product is 5 × 3 × 1 = 15, and so on. The number of possible
trees increases explosively as the number of species goes up. For 50 species, there are
about 3 × 10 76 possible unrooted trees, and for the 30 million species that may be alive
on Earth today the number is about 10 300,000,000 . No computer can search through that
quantity of trees and about 25 or so species is the practical upper limit.
Students of molecular phylogenies distinguish between “algorithms” and “optimality
criteria.” Maximum likelihood and parsimony are examples of optimality criteria,
which say that the best tree is the one requiring the least evolutionary change. An
optimality criterion is a criterion that all the possible phylogenies can be compared
against, and the best estimate of the phylogeny is the one that is closest to the criterion. 5
Optimality criteria run into the problem of limited computer search capacity, because
all the trees have to be compared with the criterion. If the number of species is too big
for all the possible trees to be searched, the search instead has to be done by means of an
“algorithm.” An algorithm is a rule about how to search from one tree to the next, and
to assess which of the two trees is better. It will eventually find a tree that is better than
any of the alternatives it compares it with, but it searches through only a limited number
of trees to reach that end.
Here is an analogy. Suppose you are in San Francisco and giving someone instructions
on how to find Los Angeles. An optimality criterion would be to say “find the city
with the largest population in the USA.” The unfortunate person who receives this
direction has to visit every city in the country, and measure their population sizes, in
order to be sure he or she has found that destination. (We assume they have no other
source of information.) An algorithm would be something like “face south and, keeping
the Pacific Ocean on your right-hand side, move forwards until you arrive at a city
with more than a million inhabitants.” Now only a small proportion of the USA has to
be searched, and the conclusion will be satisfactory so long as no other cities exist that
meet the criterion between the starting and finishing points.
The particular algorithms used in phylogenetic research have constantly improved
in recent years, and we shall not enter into details here. What does matter is that
5 We could say, formally, that the optimality criterion of parsimony is zero evolutionary change: the tree of
all the possible trees that comes closest to having zero change is the best. Notice that is not the same as saying
we expect any tree to have zero change: we know that evolution has happened. It is a formal logical criterion,
not a theory of reality.