On the Alexander dual of the path ideals of rooted and unrooted trees

Let R be a commutative Noetherian ring and I be an ideal of R. We say that I satisfies the persistence property if Ass(R) (R/I-k) subset of Ass(R) (R/Ik+1) for all positive integers k >= 1, where Ass(R)(R/I) denotes the set of associated prime ideals of I. In this paper, we introduce some classes of square-free monomial ideals in the polynomial ring R = K[x(1), ... , x(n)] over a field K which are associated to rooted and unrooted trees. In fact, we show that the Alexander dual of the monomial ideal generated by the paths of maximal lengths in an unrooted starlike tree (respectively, a rooted starlike tree) has the persistence property (respectively, is normally torsion-free).