a species tree topology $\psi$ on $\mathcal X$, so that if $\mathcal A\subseteq \mathcal X$, then $\mathcal A\subseteq {\operatorname{desc}_\mathcal X}({\operatorname{MRCA}}(\mathcal A))\subseteq \mathcal
  X$. A *clade* $\mathcal C$ on $\psi$ is a subset of $\mathcal
  X$ such that $\mathcal C={\operatorname{desc}_\mathcal X}({\operatorname{MRCA}}(\mathcal C)).$

The notions of MRCA and clade extend to gene trees in an obvious way, replacing $\mathcal X$, $\psi$, $\mathcal C$, with $\mathcal X_g$, $T$, $\mathcal C_g$ in the definitions.

For a gene tree $T$, the set of all clades on $T$ is denoted $\mathcal H(T)$. Similarly, for a species tree $\sigma=(\psi,\lambda)$ the set of clades on $\psi$ is denoted $\mathcal H(\sigma)=\mathcal H(\psi)$.

In discussing the relationships between a subset $\mathcal Y$ of the taxa $\mathcal X$ on a tree $\psi$, we use the terminology of a *displayed tree*: a tree obtained from the full tree by first passing to the rooted subtree spanned by $\mathcal Y$, and then suppressing any non-root nodes of degree 2 [@semple2003]. As an example, the species tree in Fig. \[F:partition\] displays $((b,d),e)$. The notion of displayed trees can be applied in the context of either species trees (with or without branch lengths) or gene trees.

A detailed presentation of the multispecies coalescent model is 