A Simple Solution?

The trivial solution to the phylogeny problem would be to enumerate over all possible trees and calculate the target function for each one. However, the number of non-isomorphic, labeled, binary, rooted trees containing n leaves, can be shown to be:

$$(2n-3)!! = \prod_{i=2}^{n}(2i - 3)$$ (1)

(or (2n-5)!! for unrooted trees) which is of course super-exponential - for n=20, for instance, there are about 10²¹ such trees. This means that exhaustive enumeration is unfeasible even for a relatively small number of species.

The next sections will present several approaches towards defining a target function, and attempting to solve the problem for that target function.

  

Figure 8.1: A most parsimonious 5-species phylogeny for a single DNA site. The bars mark the two possible edges along which a mutation might have occurred.

  

Figure 8.2: The unrooted counterpart of the phylogeny in figure 8.1. Notice that there is now no ambiguity about the placement of the mutation.