is displayed on $\sigma$. Thus $c \notin {\operatorname{desc}_\mathcal X}({\operatorname{MRCA}}(\mathcal C))$.

That $1/3$ cannot be replaced with a smaller number is a consequence of Corollary \[cor:third\].

Clade invariants {#sec:inv}
================

A *clade invariant* for a species tree topology is a polynomial in the probabilities of clades on gene trees that vanishes for all edge length assignments to the species tree. More completely, a clade invariant associated to an $n$-taxon species tree topology $\psi$ is a multivariate polynomial in $2^{n}-n-2$ indeterminates (one for every non-trivial clade) which evaluates to zero at any vector of clade probabilities ${\mathbb P}_\sigma(\mathcal C_g)$ arising from $\sigma=(\psi, \lambda)$, regardless of the values of $\lambda$.

Proposition \[prop:trivinv\] gives an example of a clade invariant for binary gene trees that, in addition, is independent of all features of $\psi$ except the number of taxa: $$\sum_{\mathcal C_g\subset \mathcal X_g\atop \text{non-trivial}}\mathbb P_\sigma(\mathcal C_g) - (n-2)=0.$$ We call this the *trivial invariant*, and emphasize it is satisfied by clade probabilities from any species tree on $\mathcal
X$.

Clade invariants can be computed for small trees using computational algebra software, such as Singular [@GPS09]. For each edge length $\lambda_i$, one sets $\Lambda_i=\exp(-\lambda_i)$, and then expresses the clade probabilities as multivariate polynomials in the 