Leaf-Preserving Distance between Rooted Labeled Caterpillars

In this paper, we introduce a leaf-preserving distance between rooted labeled trees (trees, for short) as a distance that a leaf in a tree is corresponding to a leaf in another tree. Then, we show that the leaf-preserving distance is always smaller than or equal to the bottom-up distance and incomparable with the alignment distance, the isolated-subtree distance and the segmental distance, but the problem of computing the leaf-preserving distance between trees is MAX SNP-hard. On the other hand, for a rooted labeled caterpillar (caterpillar, for short) that is a tree transformed to a rooted path after removing all the leaves in it, we design an algorithm to compute the leaf-preserving distance between caterpillars in O(h2λ) time, where h is the maximum height and λ is the maximum number of leaves for given two caterpillars. Finally, we give experimental results for computing the leaf-preserving distance for caterpillars.