Edit distance between unlabeled ordered trees

CNRS, LIAFA, Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France; amicheli@liafa.jussieu.fr; rossin@liafa.jussieu.fr

Received: 15 June 2005
Accepted: 15 February 2006

Abstract

There exists a bijection between one-stack sortable permutations (permutations which avoid the pattern (231)) and rooted plane trees. We define an edit distance between permutations which is consistent with the standard edit distance between trees. This one-to-one correspondence yields a polynomial algorithm for the subpermutation problem for (231) pattern-avoiding permutations. Moreover, we obtain the generating function of the edit distance between ordered unlabeled trees and some special ones. For the general case we show that the mean edit distance between a rooted plane tree and all other rooted plane trees is at least n/ln(n). Some results can be extended to labeled trees considering colored Dyck paths or, equivalently, colored one-stack sortable permutations.