As tree-adjoining grammars operate with trees, I suppose they can be considered as a kind of tree grammars. If this assumption is correct, I'm wondering: where should we place them in the tree grammar hierarchy?
By the tree grammar hierarchy I mean one which is similar to the Chomsky hierarchy of word grammars, e.g. as on this figure taken from
Matsuda K., Mu SC., Hu Z., Takeichi M. (2010) A Grammar-Based Approach to Invertible Programs. In: Gordon A.D. (eds) Programming Languages and Systems. ESOP 2010. Lecture Notes in Computer Science, vol 6012. Springer, Berlin, Heidelberg