A distance metric on binary trees using lattice-theoretic measures

In this paper, we propose a new metric for rooted trees with unlabeled vertices based on alignments of nested parenthesis strings. We prove that the time complexity for computing this metric is NP-hard and present a 1.5-approximation algorithm for it.

We develop combinatorial methods for establishing lower bounds on the rotation distance between binary trees, i.e., equivalently, on the flip distance between triangulations of a polygon. These methods lead to sharp estimates for certain particular pairs of trees. As an application, we prove that, for each n, there exist size n trees at distance $2n-O(\sqrt{n})$, i.e., the diameter of the nth associahedron has at least this value.

We present an analysis of two transformations on well-formed parentheses strings. Using a lattice approach, the corresponding least-move distances are computable, the first in linear time and the second in quadratic time.

We consider a transformation on rooted unlabeled binary trees which is a special instance of the well-known general rotation. Using lattice-theoretic results, we give an efficient algorithm for computing the shortest path distance between two binary trees.