Well-quasiordering finite trees with gap-condition. Proof of Harvey Friedman's conjecture

Let s, t be trees with edges labelled by ordinals. We say that s is less-or-equal to t with respect to the gap-conditon if s can be embedded into t so that each edge is mapped onto a path consisting of edges with greater-or-equal labels. We show that finite trees are well-quasiordered with respect to the gap-condition. This solves a problem posed by Harvey Friedman.