lengths of pendant edges, since only the lengths of internal edges affect probabilities of gene tree topologies under the multispecies coalescent. For example, we write $((a,b){\text{:}}t, c)$ for a 3-taxon species tree with one internal edge with length $t$, measured in coalescent units. If there is a constant effective population size, $N$, over an edge of the species tree, then a length of $t$ indicates that the edge represents $Nt$ generations [@degnan2009]. For varying effective population size, a non-linear scaling is needed to relate coalescent units to generations. Species trees are thus not assumed to be ultrametric in coalescent units.

In discussing trees, we find it convenient in various settings to use either spatial or temporal terminology. For instance, if $(v,w)$ is a directed edge in $\psi$ pointing away from the root, then we may say that $v$ is *above*, or an *ancestor*, of $w$ and that $w$ is *below*, or a *descendant*, of $v$. Natural extensions of these terms should be clear from context.

We denote taxa in $\mathcal X$ by lower case letters such as $a,b,c,\dots$. To distinguish between taxa and sampled genes from those taxa, we use the corresponding upper case letters $A,B,C,\dots$ to denote the genes, 