$$\label{E:trivinv}
\sum_{\mathcal C_g \subset \mathcal X_g\atop\text{non-trivial}} \mathbb P(\mathcal C_g)=\sum _{T} (n-2) \mathbb P(T)
=n-2.$$

\[thm:toomany\] For an arbitrary distribution of binary gene trees on a taxon set $\mathcal
  X$ with $|\mathcal X|\ge 4$, the gene tree probabilities ${\mathbb P}(T)$ cannot be identified from the clade probabilities ${\mathbb P}(\mathcal
  C_g)$.

The set $\mathcal X_g$ has $2^n-n-2$ subsets $\mathcal C_g$ with $2\le
  |\mathcal C_g|\le n-1$. Using Proposition \[prop:trivinv\], the clade probabilities can thus be specified by point in a $(2^n-n-3)$-dimensional vector space. However, there are $(2n-3)!!=1\cdot 3\cdots(2n-3)$ binary gene trees on $\mathcal X_g$, so a gene tree distribution is specified by a point in a $((2n-3)!!-1)$-dimensional vector space. But since $$(2n-3)!!-1>2^n-n-3$$ when $n\ge 4$, and the map from gene tree probabilities to clade probabilities is linear, the map is not invertible at any point.

We note that for an arbitrary distribution on multifurcating gene tree topologies, the trivial invariant in Eq.  need not hold. However, the argument establishing Theorem \[thm:toomany\] can be modified to apply to such distributions, since the number of multifurcating trees is greater than the number of binary ones.

Highly probable gene tree clades are species tree clades {#sec:high}
========================================================

For the remainder of the paper, we assume that 