A Gap Tree Theorem for Quasi-Ordered Labels

Given a quasi-ordering of labels, a labelled ordered tree s is embedded with gaps in another tree t if there is an injection from the nodes of s into those of t that maps each edge in s to a unique disjoint path in t with greater-or-equivalent labels, and which preserves the order of children. We show that finite trees are well-quasiordered with respect to gap embedding when labels are taken from an arbitrary well-quasi-ordering provided every tree path can be partitioned into a bounded number of subpaths of comparable nodes. This extends Kř́ıž’s combinatorial result [7] to partially ordered labels.