## PhylogeneticsPhylogenetics' is the reconstruction and analysis of phylogenetic (evolutionary) trees and networks based on inherited characteristics. It is a flourishing area of intereaction between mathematics, statistics, computer science and biology. The main role of phylogenetic techniques lies in evolutionary biology, where it is used to infer historical relationships between species. However, the methods are also relevant to a diverse range of fields including epidemiology, ecology, medicine,as well as linguistics and cognitive psychology This graduate-level book, based on the authors lectures at The University of Canterbury, New Zealand, focuses on the mathematical aspects of phylogenetics. It brings together the central results of the field (providing proofs of the main theorem), outlines their biological significance,and indicates how algorithms may be derived. The presentation is self-contained and relies on discrete mathematics with some probability theory. A set of exercises and at least one specialist topic ends each chapter. This book is intended for biologists interested in the mathematical theory behind phylogenetic methods, and for mathematicians, statisticians, and computer scientists eager to learn about this emerging area of discrete mathematics. 'Phylogenetics' in the 24th volume in the Oxford Lecture Series in Mathematics and its Applications. This series contains short books suitable for graduate students and researchers who want a well-writtenaccount of mathematics that is fundamental to current to research. The series emphasises future directions of research and focuses on genuine applications of mathematics to finance, engineering and the physical and biological sciences. |

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### Contents

Preliminaries | 1 |

Trees and splits | 43 |

Compatibility of characters | 65 |

Maximum parsimony | 84 |

Subtrees and supertrees | 110 |

Treebased metrics | 145 |

Markov models on trees | 183 |

References | 218 |

Commonly used symbols | 231 |

### Common terms and phrases

bijection binary characters binary phylogenetic tree character distribution chordal graph completes the proof construct convex Corollary corresponding defined definitive denote the set describe digraph dissimilarity map distinct elements easily checked equivalence Example excess-free follows four-point condition full character function Furthermore graph G induced int(C interior edge interior vertex intersection graph interval graph isomorphic label set leaf labelled Lemma Markov process matrix maximal maximum parsimony maximum parsimony tree metric space minimum extension NP-complete pair pairwise compatible parsimony score path polynomial polynomial-time algorithm probability problem proof of Theorem Proposition quartet trees RB(n restricted chordal completion result rooted binary phylogenetic rooted phylogenetic tree rooted tree rooted X-tree semi-labelled tree set of X-splits shown in Fig species splits Splits-Equivalence Theorem SPR operation stationary process subgraph subset Suppose Theorem tree and let tree metric representation tree shape ultrametric unrooted vertex set vertices X-tree x,y e