The following is based on a weekend project that I also presented as a short talk in an undergraduate combinatorics seminar. The project is self-contained and mostly based on independent work. Ideas and inspiration came from discussions with my teacher and from the introduction of Diaconis and Holmes (1998). Theorem 2 is from Semple and Steel (2003). Tree pictures were produced with Sagemath and Latex.
A phylogenetic tree is a rooted binary tree with labeled leaves.
These trees are used in biology to represent the evolutive history of species. The leaves are the identified species, the root is a common anscestor, and branching represents speciation.
An interesting problem is that of reconstructing the phylogenetic tree that best explains the observed biological characterics of a set of species. A naive mathematical formulation of this problem is proposed in section 4, and used to implement a tree reconstruction algorithm.