Before revision: This note introduces the exact solver Bute for the exact treedepth problem, along with two variants of the solver . Each of these solvers computes an elimination tree  in a bottom-up fashion by finding sets of vertices that induce subgraphs of small treedepth, then combining sets of vertices together with a root vertex to produce larger sets. The algorithms make use of a trie data structure to reduce the effort required to determine acceptable combinations of subtrees.
After revision: This note introduces the exact solver Bute for the exact treedepth problem, along with two variants of the solver . Each of these solvers computes an elimination tree in a bottom-up fashion . Three features greatly improve the algorithm's run time. The first of these is a specialised trie data structure to reduce the effort required to determine acceptable combinations of subtrees.