A note on partially ordered tree automata

A recent paper by Bouajjani, Muscholl and Touili shows that the class of languages accepted by partially ordered word automata (or equivalently accepted by Σ2-formulae) is closed under semi-commutation and it suggested the following open question: can we extend this result to tree languages? This problem can be addressed by proving (1) that the class of tree regular languages accepted by Σ2 formulae is strictly included in the class of languages accepted by partially ordered automata, and (2) that Bouajjani and the others results cannot be extended to tree.