some edge of $T$ represents $C$. We let $Cl(\mathcal{T}) = \cup_{T \in \mathcal{T}} Cl(T)$. Whenever we (reasonably) assume that all singleton clusters are present in the input[^4], it is easy to see that every cluster set $\mathcal{C}$ on ${{\mathcal X}}$ can be written as $Cl(\mathcal{T})$ for some $\mathcal{T}$ as follows. We take any proper coloring of $IG(\mathcal{C})$ (i.e. map the nodes of $IG({{\mathcal C}})$ to colors such that no two adjacent nodes have the same color) and use the resulting colors to partition $\mathcal{C}$. Clusters that have been colored the same are all mutually compatible, so can be represented by a single tree corresponding to that color. Finally, whenever a subset of clusters pertaining to a color does not cover all elements of $\mathcal{X}$, the missing taxa can be attached to the root. An obvious corollary of this is that the chromatic number of $IG(\mathcal{C})$ is a lower bound on the cardinality of $\mathcal{T}$.

Whenever ${{\mathcal C}}=Cl(\mathcal{T})$ there is an important relationship between the nodes and edges of trees in $\mathcal{T}$, and the (maximal) ST-sets of $\mathcal{C}$. Let $T$ be a (not necessarily binary) tree on ${{\mathcal X}}$. In Section \[sec:prelim\] we defined when an edge of a tree represents 