A central and challenging problem in contemporary biology is how to
accurately reconstruct evolutionary trees from DNA sequence data.
This thesis addresses three themes from this endeavour -- comparison,
consistency and confidence intervals -- by analysing distributions arising
from phylogenetic trees.
Toward the first theme, the distribution of the symmetric difference
metric on pairs of binary and phylogenetic trees is studied, and a number
of new results obtained. These theorems, as well as a result on another
tree metric answer previous conjectures in this area. Also under the
theme of comparison, we analyse distributions on bicoloured trees arising
from the principle of parsimony. A streamlined proof is given of an
elegant theorem which allows an efficient comparison of how much better
a maximum parsimony tree fits given data than a randomly-chosen tree. A
dual distribution, where the tree is fixed and the data varies is also
analysed, answering a recent unsolved problem.
We then consider the theoretical accuracy of tree-building methods,
concentrating on the statistical property of consistency. Under a simple
stochastic model on bicoloured trees, conditions for the consistency of
frequently-used methods based on parsimony and compatibility are
examined. lt is shown that even in "best possible" conditions both
methods can be inconsistent, though a strong sufficient condition for
compatibility is given. The analysis is extended for a molecular clock.
Finally, procedures are described for placing confidence intervals around
phylogenies, and limitations on the sort of confidence intervals possible
are given. Ways to efficiently implement these procedures are then
considered -- in particular, approximate methods, applications to sets of
taxa of size four, and simplifications under a molecular clock.
The rate that sequence data must grow as a function of the number of taxa
for confidence intervals to converge to a single tree is also considered.
The arguments in this thesis are primarily combinatorial and stochastic.
In the hope that their implications will also interest biologists, some
space has been given to motivating and explaining the biological relevance
of the results presented.