Dealing with non-tree like data: biological examples and problems

Rooted phylogenetic networks: brief overview of rooted networks with examples from literature and why they are of interest. Some ideas for how to obtain rooted phylogenetic networks from clusters, trees, multi-labelled trees, triples (including brief discussion on polynomical vs NP-complete problems).

Metrics, quartet systems and split systems => X-trees

Methods to get trees from metrics (e.g. Buneman tree and refined Buneman tree), from split systems (e.g. tree-popping), and quartet-systems (e.g. quartet-joining).

X-nets from arbitrary metrics and split systems

Split networks and how to obtain them using split-decomposition and neighbornets. Introduction to median and quasimedian neworks, and brief mention of their link with tight-spans.

Measuring and removing inconsistencies

Consistency of tree/network building methods and desirable properties of such methods (illustrated by, e.g. Buneman trees). Measures of inconsistency, including k-compatibility and Delta-hyperbolicity, and how they can lead to interesting combinatorial results and be exploited for real data.