Alternating tree automata for arbitrary arity tree 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):
An alternating tree automaton for arbitrary degree trees has a transition function of the following type:
$$\delta:Q\times \Sigma\times D\to {\cal B}(\mathbb{N}\times Q)$$ where ${\cal B}$ is the set of Boolean functions over the given set.

This has a limitation that $\delta(q,\sigma,d)$ only outputs values between $1,...,d$ (so it fits the degree of the tree).

Thus, the transition function is defined separately for each degree, so it's effectively a family of automata. In particular, this means that if you want to handle arbitrary degrees, you need an infinite description (or a finite description of an infinite object). While this is mathematically possible, it can of course make things undecidable, if (for example) your transition function is given by a TM.

In your example, you can clearly construct such an automaton, since you can construct one for every degree.