Title:

Subtree transfer operations on phylogenetic networks

Leaflabelled, rooted binary trees can be used to represent a wide range of data. One way to measure the discrepancy between a pair of such trees, is to measure how many subtree transfer operations are required to transform one into the other. Four standard subtree transfer oper ations for binary trees, used in particular for leaflabelled binary trees, are: tree bisection and reconnection (TBR), subtree prune and regraft (rSPR), rooted subtree prune and regraft (rSPR) and nearest neighbour interchange (NNI). Each of these operations induces a metric over the class, B(n), of binary trees with leaves labelled from 0 to nâ1. For a pair of trees in B(n), we investigate the maximum possible distance between them, as well as the typical distance if they were chosen randomly from some exchangeable distribution. The TBR (and rSPR) distances between two trees are equivalent to the size of a minimal unrooted agreement forest (and minimal rooted agreement forest respectively) of the pair of trees. The notion of an agreement forest generalises to arbitrarily large sets of trees. For a fixed integer, k, we determine quite tight bounds on the worstcase size of an agreement forest for a set of k trees in B(n). We also estimate the expected size of a minimal agreement forest of k randomly chosen trees, each independently chosen from their own exchangeable distribution. We then construct two network operations: the phylogeny subtree prune and regraft (pSPR) and the phylogeny nearest neighbour interchange (pNNI), which are generalisations of the rSPR and NNI operations, respectively. The pSPR and pNNI, which act over the class of phylogenies (phylogenetic networks with recombination), are both special cases of the edgeexchange (EEX) operation. We investigate the conditions under which it is possible to transform any phylogeny into any other phylogeny, and present some bounds on the number of operations required. We also consider the extreme phylogenetic triple problem: For a given integer, n â¥ 3, what is the minimum size of a set of rooted binary trees, with leaves labelled 1 to n, such that every phylogenetic triplet, (abc), is satisfied by at least one tree in the set? We show that the answer to this question is Î(log log n), by relating it to the nonbetweenness problem. The final two chapters focus on two problems in communication network theory: the multiple unicast problem (the guessing game) and the information flow problem. For a fixed number of colours, s, we determine the guessing number of all large enough cycles, which answers a longstanding open question. We also provide a detailed constructive solution to the information flow problem on the full clock digraph, and on clock digraphs in general. These results solve two open problems, both had been conjectured 8 years ago.
