display the rooted triple $((a,b),c)$. Thus, the rooted triple probability satisfies $\mathbb P_{\sigma}( ((A,B),C ))< (1/3)
  \exp(-\epsilon)$ [@nei1987]. But then $$(1/3)
  \exp(-\epsilon)>\mathbb P_{\sigma} (((A,B),C)) \ge \mathbb
  P_{\sigma}(\mathcal C_g),$$ since $((A,B),C)$ is displayed on every gene tree on which $\mathcal C_g$ is a clade.

To establish the last claim of the theorem, we construct an example. For any set $\mathcal C$ with $1<|\mathcal C|<|\mathcal X|$, pick some $a\in \mathcal C$, and some $c\in \mathcal X\smallsetminus \mathcal
C$. Let $\mathcal C'=\mathcal C\smallsetminus\{a\}$. Consider a binary species tree $\sigma$ which has a subtree of the form $((a,c){\text{:}}\delta,T_{\mathcal C'}{\text{:}}\gamma)$, where $T_{\mathcal C'}$ is any rooted tree on $\mathcal C'$. Note then that $\mathcal C$ is not a clade on $\sigma$.

By taking $\gamma$ to be large, the probability that the lineages from $\mathcal
C'$ coalesce below ${\operatorname{MRCA}}(\{a,c\}\cup\mathcal C')$ can be made as close to 1 as desired. Because the probability that lineages $A$ and $C$ fail to coalesce within time $\delta$ is $\exp(-\delta)$, by also choosing $\delta\approx\epsilon$ the probability that three lineages (one for $A$, one for $C$, and one for $\mathcal C_g'$) enter the ancestral population above this MRCA can be made as close to $\exp(-\epsilon)$ as we wish. Thus the probability that 