trees. We call ${\ensuremath{\mathcal{P}}}$ a *profile*. We denote $\bigcup_{T \in {\ensuremath{\mathcal{P}}}}{\ensuremath{\mathcal{L}}}(T)$ by ${\ensuremath{\mathcal{L}}}({\ensuremath{\mathcal{P}}})$. A *supertree* of ${\ensuremath{\mathcal{P}}}$ is a phylogenetic tree whose label set is ${\ensuremath{\mathcal{L}}}({\ensuremath{\mathcal{P}}})$.

Let $S$ be a phylogenetic tree. For any $Y \subseteq {\ensuremath{\mathcal{L}}}(S)$ let $S|Y$ denote the tree obtained by suppressing any degree two vertices in the minimal subtree of $S$ connecting the labels in $Y$. Let $T$ be a phylogenetic tree where ${\ensuremath{\mathcal{L}}}(T) \subseteq {\ensuremath{\mathcal{L}}}(S)$. We say that $S$ *displays* $T$ if $T$ can be derived from $S|{\ensuremath{\mathcal{L}}}(T)$ by contraction of edges. Given a profile ${\ensuremath{\mathcal{P}}}$ of trees, the *tree compatibility* problem asks if there exists a supertree of ${\ensuremath{\mathcal{P}}}$ which displays all the trees in ${\ensuremath{\mathcal{P}}}$. If such a supertree $S$ exists, we say that ${\ensuremath{\mathcal{P}}}$ is compatible and $S$ is a compatible tree of ${\ensuremath{\mathcal{P}}}$. The tree compatibility problem is NP-complete [@Steel92] but fixed parameter tractable when parametrized by number of trees [@BryantLagergren06].

An instance of the tree compatibility problem can be transformed to an instance of character compatibility by representing each input tree by its set of splits [@SempleSteel03]. A *quartet* is a binary phylogenetic tree with exactly four leaves. An instance of tree compatibility can also be transformed into an instance 