Combinatorial statistics on phylogenetic trees

The analysis of phylogenetic trees is important when trying to reconstruct the evolutionary history of sets of species and infer speciation events. In this thesis we evaluate several combinatorial statistics on classes of phylogenetic trees using the leaf and vertex generating functions as our basic tools. We derive the vertex and leaf generating functions via a simple tree decomposition method that distinguishes a vertex and then counts all trees on a fixed number of edges with this distinguished vertex. Using these basic generating functions, and a decomposition method via the least common ancestor we derive formulas for n-element sets of leaves and anti-chains. Using Riordan matrices and the Fundamental Theorem of Riordan Arrays we can count the total number of leaf sets and the total number of anti-chains. Martin Klazar's gap extension method is translated into a Riordan matrix context and we reproduce and extend his result on anti-chains in ordered trees.