User blog comment:Hyp cos/extended Kruskal theorem/@comment-25601061-20180604152447/@comment-11227630-20180609005833

No. 1-labeled trees have order type \(\vartheta(\Omega^\omega)\), 2-labeled trees have order type \(\vartheta(\Omega_2^\omega+\vartheta_1(\Omega_2^\omega)^\omega)\). Generally the order type of n-labeled trees is slightly above \(\vartheta(\Omega_n^\omega)\), but when they are union together, their supremum, or the growth rate of wgap function, is still \(\vartheta(\Omega_\omega)\).