  This thesis develops and expands upon known techniques of mathematical
physics relevant to the analysis of the popular Markov model of phylogenetic
trees required in biology to reconstruct the evolutionary relationships of
taxonomic units from biomolecular sequence data. The techniques of mathematical
physics are plethora and have been developed for some time. The Markov model of
phylogenetics and its analysis is a relatively new technique where most
progress to date has been achieved by using discrete mathematics. This thesis
takes a group theoretical approach to the problem by beginning with a
remarkable mathematical parallel to the process of scattering in particle
physics. This is shown to equate to branching events in the evolutionary
history of molecular units. The major technical result of this thesis is the
derivation of existence proofs and computational techniques for calculating
polynomial group invariant functions on a multi-linear space where the group
action is that relevant to a Markovian time evolution. The practical results of
this thesis are an extended analysis of the use of invariant functions in
distance based methods and the presentation of a new reconstruction technique
for quartet trees which is consistent with the most general Markov model of
sequence evolution.
