Alternating tree automata for arbitrary arity tree
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.