Olivier Finkel and Stevo Todorcevic
We prove that the injectively ω-tree-automatic ordinals are the ordinals smaller than ωωω . Then we show that the injectively ωn-automatic ordinals, where n ≥ 1 is an integer, are the ordinals smaller than ωωn. This strengthens a recent result of Schlicht and Stephan who considered in  the subclasses of finite word ωn-automatic ordinals. As a by-product we obtain that the hierarchy of injectively ωn-automatic structures, n ≥ 1, which was considered in , is strict.
Keywords: ω-tree-automatic structures; ωn-automatic structures; ordinals