Could alternating tree automata be used for recognizing set (language) of arbitrary-arity trees?
More specifically, as an example: let $\Sigma = \{a,b,c\}$ - labels for tree nodes. Trees from $T$ are of the form: from each node only one of the following children could appear: $\{a,b\},\{b\},\{c\}$.
I.e. tree will be of the form, smth like this ($e$ on picture is just root of the tree, indexes on labels are just features of drawing tool, they aren't related to the question):