\delta_i$ have a common parent in $\psi^*$.

The *pruned species tree topology* $\psi$ on $\mathcal X$ is obtained from $ \psi^*$ by labeling the parent of the $x_{ij}$ by $x_i$ for each $i$ with $\delta_i > 1$, and then excising the leaves $x_{ij}$ and the pendant edges on which they lie.

Note that while an extended species tree gives rise to a species tree by the pruning process, in an extended species tree a branch length is assigned to those edges which become pendant in the species tree whenever there are two or more sampled individuals in the taxon. Since our notion of a species tree in this paper does not have pendant edge lengths, an extended species tree thus carries more edge length information than the associated species tree.

Gene trees arising from the coalescent model on an extended species tree have leaves labelled $\mathcal X_g = \{X_{11}, \dots, X_{1\delta_1},
\dots, X_{n1}, \dots, X_{n\delta_n}\}$ and are, with probability 1, binary. One readily checks that the probability of such a gene tree under the multispecies coalescent on the extended species tree is exactly the same as the probability of the gene tree under a multiple individual sampling scheme on the 