Comparing trees via crossing minimization

We consider the following problem (and variants thereof) that has important applications in the construction and evaluation of phylogenetic trees: Two rooted unordered binary trees with the same number of leaves have to be embedded in two layers in the plane such that the leaves are aligned in two adjacent layers. Additional matching edges between the leaves give a one-to-one correspondence between pairs of leaves of the different trees. Our goal is to find two planar embeddings of the two trees (drawn without crossings) that minimize the number of crossings of the matching edges. We derive both (classical) complexity results and (parameterized) algorithms for this problem (and some variants thereof).1