# Summer School on Phylogenetic Combinatorics - Lectures A. Dress

Phylogenetic combinatorics is a branch of discrete applied mathematics concerned with providing and analysing combinatotrial tools for phylogenetic "taxonomy" (according to WIKIPEDIA "the academic discipline of defining groups of biological organisms", see ). As the final result of a taxonomic analysis will, in general, be presented in terms of a phylogenetic tree or sometimes, in more "advanced" form (leaving room for doubts), in terms of a phylogenetic network, such trees and various related mathematical structures such as "X-nets" and "tight spans" will, therefore, constitute the main topic to be discussed in this series of 6 one-hour lectures. The lectures will aim to illustrate that these biologically-inspired concepts provide an abundant reservoir of interesting -- and sometimes also quite challenging -- combinatorial problems and algorithmic tasks worth to be studied not only in view of their sometimes quite fascinating practical applications in phylogeny, but also in view of their intrinsic mathematical appeal and interest.
More specifically, there are (Dress, A. et al, Basic Phylogenetic Combinatorics, Cambridge University Press. 2012) three principal options for encoding phylogenetic trees: split systems, quartet systems, and metrics. Such encodings provide useful options for analyzing and manipulating phylogenetic trees and networks, and they are at the basis of much of phylogenetic data processing. In the lectures, it will be explained how each of these three encodings provides a unique perspective for viewing, perceiving, and analysing the combinatorial structure of a phylogenetic tree and how they are interrelated (see the figure).
In the first lecture, X-trees and X-networks will be introduced and biologically relevant examples will be provided. Next, it will be described how X-trees give rise to metrics, quartet systems and split systems over X and those quartet or split systems or metrics over X that correspond to an X-tree will be characterized in terms of simple combinatorial conditions (Lecture 2 and 3). And then, as "naturally" obtained quartet or split systems or metrics rarely satisfy these conditions, the problem of how to deal best with data that do not simply correspond to an X-tree will be addressed (Lecture 4 to 6).