Dominator Tree for DAG
Recall the definition of dominators:

$$\hbox{Dom}(n_o) = \{ n_o \}$$
$$\hbox{Dom}(n) = \{ n \} \cup \bigcap_{p \in \hbox{preds}(n)} \hbox{Dom}(p)$$

This can be done in a single pass over all nodes, if you can guarantee that you visit all predecessors of a node before you visit that node. So perform a topological sort, and you're done.

It may be possible to calculate dominator trees directly (i.e. without first computing dominators) if the graph is a DAG, but I'm not certain how you would do this.