Phylogenetic Tree Inferences Using Quartet Splits

Phylogenetic trees are used by evolutionary biologists and geneticists as a way of classifying the relationships between different taxonomic units (i.e. species, subspecies, genes). The branching of a tree represents one taxonomic unit evolving into two or more different units. These events are called speciation events. The leaves of phylogenetic trees are labeled with the taxa that are to be studied. For any tree, and phylogenetic trees in particular, deleting an edge results in a bipartion of the leaf label set. We call such deletions {underline{splits}} and represent the split that results in bipartion sets $A$ and $B$ by $A|B$. A split $\sigma$ is realized by a tree $T$ if some combination of edge deletions and edge contractions of $T$ results in the bipartition expressed by $\sigma$. The set of all splits realized by $T$ is denoted $\Sigma (T)$. We discuss what it means for two splits to be compatible and give a proof of the Splits-Equivalence Theorem which states that for a collection $\Sigma$ of splits, there is a tree $T$ such that $\Sigma=\Sigma(T)$ if and only if the splits of $\Sigma$ are pairwise compatible. Of particular interest are splits in which both $|A|$ and $|B|=2$. These are referred to as {\underline{quartet splits}}. We say that a set $\A$ of quartet splits infers another quartet $s\notin \A$ if every tree that displays $\A$ must also display $s$. In his Master\u27s Thesis {\textit{Reconstruction Methods for Derivation Trees}}, Dekker showed that for any $r$, there are $r$-ary inferences that can be made from a set of $r$ quartet splits but cannot be made from fewer. We present an independent recreation of a portion of Dekker\u27s work by giving an exhaustive list of all binary quartet split inferences, as well as show that there are no ternary inferences that can be made when the number of leaves is greater than 8. We also provide an example of a valid ternary quartet split inference