Gap embedding for well-quasi-orderings

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-quasi-ordered with respect to gap embedding when labels are taken from an arbitrary well-quasi-ordering such that each tree path can be partitioned into a bounded number of subpaths of comparable nodes. This extends Krǐž's result and is also optimal in the sense that unbounded incomparability yields a counterexample.