Phylogenetic networks generalize evolutionary trees and are commonly used to represent evolutionary relationships between species that undergo reticulate evolutionary processes such as hybridization, recombination and lateral gene transfer. In this thesis all quarnets, networks on four species, of a network are assumed to be known. We prove that each recoverable undirected or semi-directed binary level-2 phylogenetic network without redundant biconnected components is encoded by its set of quarnets, meaning that the network is uniquely determined by its quarnets. Furthermore, two decomposition theorems for undirected and semi-directed binary phylogenetic networks are presented. These decomposition theorems are proved for undirected binary phylogenetic networks for all levels and for semi-directed binary phylogenetic networks that are at most level-2.

Phylogenetic networks generalize evolutionary trees and are commonly used to represent evolutionary relationships between species that undergo reticulate processes such as hybridization, recombination and lateral gene transfer. Recently, there has been great interest in knowing which networks are determined or encoded by their trinets, that are rooted networks on three species. Van Iersel and Moulton showed that recoverable rooted binary level-2 phylogenetic networks are encoded by their trinets. Based on their work for level-2 networks, we show here that not all recoverable rooted binary level-3 networks are weakly encoded by their trinets, but most networks are. Further, although not all level-3 networks are weakly encoded by their trinets, we are able to prove that all recoverable rooted binary level-3 networks are encoded by their quarnets, that are rooted networks on four species.