@phdthesis{Wicke2020,
  author    = {Kristina Wicke},
  title     = {Novel Aspects of Mathematical Phylogenetics},
  journal    = {Neue Aspekte mathematischer Phylogenetik},
  url       = {https://nbn-resolving.org/urn:nbn:de:gbv:9-opus-38827},
  pages     = {224},
  year      = {2020},
  abstract  = {Mathematical phylogenetics provides the theoretical framework for the reconstruction and analysis of phylogenetic trees and networks. The underlying theory is based on various mathematical disciplines, ranging from graph theory to probability theory. In this thesis, we take a mostly combinatorial and graph-theoretical position and study different problems concerning phylogenetic trees and networks. We start by considering phylogenetic diversity indices that rank species for conservation. Two such indices for rooted trees are the Fair Proportion index and the Equal Splits index, and we analyze how different they can be from each other and under which circumstances they coincide. Moreover, we define and investigate analogues of these indices for unrooted trees. Subsequently, we study the Shapley value of unrooted trees, another popular phylogenetic diversity index. We show that it may fail as a prioritization criterion in biodiversity conservation and is outcompeted by an existing greedy approach. Afterwards, we leave the biodiversity setting and consider the Shapley value as a tree reconstruction tool. Here, we show that non-isomorphic trees may have permutation-equivalent Shapley transformation matrices and identical Shapley values, implying that the Shapley value cannot reliably be employed in tree reconstruction. In addition to phylogenetic diversity indices, another class of indices frequently discussed in mathematical phylogenetics, is the class of balance indices. In this thesis, we study one of the oldest and most popular of them, namely the Colless index for rooted binary trees. We focus on its extremal values and analyze both its maximum and minimum values as well as the trees that achieve them. Having analyzed various questions regarding phylogenetic trees, we finally turn to phylogenetic networks. We focus on a certain class of phylogenetic networks, namely tree-based networks, and consider this class both in a rooted and in an unrooted setting. First, we prove the existence of a rooted non-binary universal tree-based network with n leaves for all positive integers n, that is, we show that there exists a rooted non-binary tree-based network with \$n\$ leaves that has every non-binary phylogenetic tree on the same leaf set as a base tree. Finally, we study unrooted tree-based networks and introduce a class of networks that are necessarily tree-based, namely edge-based networks. We show that edge-based networks are closely related to a family of graphs in classical graph theory, so-called generalized series-parallel graphs, and explore this relationship in full detail. In summary, we add new insights into existing concepts in mathematical phylogenetics, answer open questions in the literature, and introduce new concepts and approaches. In doing so, we make a small but relevant contribution to current research in mathematical phylogenetics.},
  language  = {en}
}