  In scientific visualization, scalar fields are often compared through edit
distances between their merge trees. Typical tasks include ensemble analysis,
feature tracking and symmetry or periodicity detection. Tree edit distances
represent how one tree can be transformed into another through a sequence of
simple edit operations: relabeling, insertion and deletion of nodes. In this
paper, we present a new set of edit operations working directly on the merge
tree as an geometrical or topological object: the represented operations are
deformation retractions and inverse transformations on merge trees, which
stands in contrast to other methods working on branch decomposition trees. We
present a quartic time algorithm for the new edit distance, which is branch
decomposition-independent and a metric on the set of all merge trees.
