Forests of label-increasing trees

Label-increasing trees are fully labeled rooted trees with the restriction that the labels are in increasing order on every path from the root; the best known example is the binary case—no tree with more than two branches at the root, or internal vertices of degree greater than three—extensively examined by Foata and Schutzenberger in A Survey of Combinatorial Theory. The forests without branching restrictions are enumerated by number of trees by F_n(x) = x(x + 1)…(x + n ? 1), n >1 (F₀(x) = 1), whose equivalent: F_n(x) = Y_n(xT₁,…, xT_n), F_n(1)= T_{n + 1} = n!, is readily adapted to branching restriction.