for an invariant that would hold for clades on trees of any size. Although only linear invariants are needed in the proof of identifiability, the invariants constructed for $k$-clades involve a linear combination of $2^{k-1}$ clade probabilities. These rather elaborate invariants and the inability to marginalize clade probabilities to smaller trees lead to a different flavor for the proof of species tree identifiability from clade probabilities.

Definitions {#sec:defs}
===========

Let $\mathcal X$ be a finite set, whose elements we refer to as *taxa*. A *species tree on* $ \mathcal X$ means a pair $\sigma = (\psi, \lambda)$, where $\psi$ is a rooted, topological tree whose leaves are bijectively labelled by elements of $ \mathcal X$, and $\lambda = (\lambda_1, \dots, \lambda_k)$ is a collection of lengths for the internal branches of $\psi$. We refer to $\psi$ as a *species tree topology*, and always assume all internal nodes of $\psi$ except the root have degree at least 3. If all internal nodes except the root have degree 3 and the root has degree 2, we say that $\psi$ and $\sigma$ are *binary*.

We use a modified Newick notation for species trees, as in [@adr2011], in which we do not specify the 