  We discuss a new memory-efficient depth-first algorithm and its
implementation that iterates over all elements of a finite locally branched
lattice. This algorithm can be applied to face lattices of polyhedra and to
various generalizations such as finite polyhedral complexes and subdivisions of
manifolds, extended tight spans and closed sets of matroids. Its practical
implementation is very fast compared to state-of-the-art implementations of
previously considered algorithms. Based on recent work of Bruns,
Garc\'ia-S\'anchez, O'Neill and Wilburne, we apply this algorithm to prove
Wilf's conjecture for all numerical semigroups of multiplicity 19 by iterating
through the faces of the Kunz cone and identifying the possible bad faces and
then checking that these do not yield counterexamples to Wilf's conjecture.
