Upper bounds for the likelihood of a phylogeny

Phylogenetics is the study of the evolutionary history of species. Given an observed set of data, deriving the optimal phylogenetic trees which explain the data with maximum probability remains a central problem in mathematical biology.Even when the phylogenetic tree is given, there is no efficient algorithm for determining branch lengths with optimal likelihood. This motivates the development of efficient upper bounds for the likelihood of a tree. These upper bounds can be used to assess the quality of a particular branch length optimization or to eliminate the trees which are not possible optimal solutions.The upper bounds derived in the thesis are based on the circular ordering of leaves of a phylogenetic tree T. They apply to extremely general models of evolution, with both equal and unequal base frequencies. We obtain tighter bounds for some specific models as well