We present an efficient algorithm that decides the consistency of partial descriptions of ordered trees. The constraint language of these descriptions was introduced by Cornell in computational linguistics; the constraints specify for pairs of nodes sets of admissible relative positions in an ordered tree. Cornell asked for an algorithm to find a tree structure satisfying these constraints. This computational problem generalizes the common-supertree problem studied in phylogenetic analysis, and also generalizes the network consistency problem of the so-called left-linear point algebra. We present the first polynomial time algorithm for Cornell's problem, which runs in time O (m n), where m is the number of constraints and n the number of variables in the constraint. © 2006 Elsevier B.V. All rights reserved.