Refined Buneman Trees

efined Buneman algorithm which is the main focus of this work belongs in this
subclass of evolutionary tree methods. The hope is that the refined Buneman
method will be less safe than its namesake and infer more splits, while still
maintaining a high degree of confidence in the splits it infers.
4.2 Parsimony methods
The foundation for this class of methods is the phylosophy of William of Ockham:
Pluralitas non est ponenda sine neccesitate — meaning something along
the lines of the best hypothesis is the one that requires the smallest number of
assumptions 2 .
This philosophy is also known as Ockham’s Razor or the parsimony principle.
In our context we shall use it to create a condition of optimality, saying that if
two proposed evolutionary processes have the same starting and ending points,
we shall assume that the simplest or shortest process is the correct one. For
example, we could imagine two substitutions of the same nucleotide working in
reverse: A → G → A — in this case we would say that no substitutions took
place at all.
The parsimony method works by considering one specific site at a time across
a set of nucleotide sequences. For each site, we postulate all possible binary tree
topologies linking these sites. We now search for a combination of assignments
of nucleotides to inner nodes such that the total number of substitutions is minimal,
selecting that topology/ those topologies for further study, since we might
not identify the same optimal topologies for all sites — we have to sum over
possible topologies to find the one that has the smallest number of substitutions
across all sites.
Figure 4.1 shows an example of estimating the number of substitutions for
a topology. Firstly, we have a tree spanning specific sites across 5 nucleotide
sequences. Secondly, we can fill in the sites for the ancestral taxa by considering
the minimum number of substitutions required, bottom up from the sites we
are already given. In this case, looking at a subtree of C and T in the lower
left corner, we know that their ancestor site must have been either C or T ,
requiring only one substitution, since all other combinations (A or G) would
require two substitutions. Thirdly, we present one solution of of many as to
how the assignment of nucleotides in ancestral sites might have been. In this
case we could make due with only two substitutions, but this solution is not
unique, there are several combinations which require only two substitutions.
The intuition for this method is quite easy to understand, and according to
[NK00] and others, under favorable conditions such as the method is expected to
produce the correct tree. However, under less favorable conditions the method is
known to produce incorrect topologies, and in any case the method is hopelessly
inefficient for large data sets, at least when using exhaustive search techniques.
[NK00] describes how search might be speeded up by using e.g. branch and
bound.
2 or even shorter: keep it simple!
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