with the set $\mathcal X_g$ denoting the full set of genes, one for each taxon. Similarly, a subset of taxa $\mathcal C \subseteq \mathcal X$ has a corresponding subset of genes $\mathcal C_g \subseteq \mathcal X_g$. A sampled gene tree from the multispecies coalescent model on $\sigma$ will thus have leaves labelled by $\mathcal X_g$, and in general may have any topology, regardless of the species tree topology $\psi$. More specifically, by a *gene tree* $T$ we mean a binary, rooted topological tree with leaves bijectively labelled by $ \mathcal
X_g$. We emphasize that for this article gene trees are topological only, with no edge lengths specified. We require that gene trees be binary, since under the multispecies coalescent only binary gene trees have positive probability.

If $\psi$ is a species tree topology on $ \mathcal X$, and $\mathcal
  A\subseteq \mathcal X$, then the *most recent common ancestor of $\mathcal A$*, ${\operatorname{MRCA}}(\mathcal A)$, is the node of $\psi$ that is ancestral to all elements of $\mathcal A$ and which is a descendant of any other node ancestral to all elements of $\mathcal A$.

Let ${\operatorname{desc}_\mathcal X}(v)\subseteq \mathcal X$ denote the elements of $\mathcal X$ descended from a node $v$ of 