Before revision:  The algorithms make use of a trie data structure to reduce the effort required to determine acceptable combinations of subtrees. Bute-Plus and Bute-Plus-Plus add a heuristic presolve step , which can quickly find a treedepth decomposition of optimal depth for many instances.
After revision:  The algorithms make use of a trie data structure . The second is a domination rule. The third is a heuristic presolve step , which can quickly find a treedepth decomposition of optimal depth for many instances.