the multispecies coalescent, the probability of any clade $\mathcal A_g \subset \mathcal X^*_g$ is the same for both the species trees in (a) and in (b).[]{data-label="F:intra"}](intraB.eps "fig:"){width=".48\textwidth"}]{} (100,150)

(2436.0, 0.0)\[l\]

$x_{11}\, x_{12}\, x_{13}\,x_{21}$ $x_{31}\, x_{41}\;\; x_{42}$ $x_{51}$

Finally, note that by construction, for each $i=1, \dots, n$, the set $\mathcal A_i = \{x_{i1},
\dots, x_{i\delta_i}\}$ is a clade on the extended species tree. But of course a set $(\mathcal A_i)_g=\{X_{i1}, \dots,
X_{i\delta_i}\}$ need not be a clade on any given gene tree.

Arbitrary gene tree distributions
=================================

Though the remainder of this paper is concerned only with the gene tree distribution arising from the multispecies coalescent model, in this section we investigate clade probabilities for arbitrary binary gene tree distributions. The main observation is that without special assumptions on the gene tree distribution, the clade probabilities do not contain enough information to recover the gene tree distribution.

Note that every gene tree must have as clades all singleton sets of gene labels, as well as the full set $\mathcal X_g$. We refer to these as *trivial clades*. Any other subset $\mathcal C_g\subset
\mathcal X_g$ is a clade on some gene trees, but not others.

For an arbitrary distribution of gene 