We define a regular tree language as in the book TATA: It is the set of trees accepted by a non-deterministic finite tree automaton (Chapter 1) or, equivalently, the set of trees generated by a regular tree grammar (Chapter 2). Both formalisms hold close resemblances to the well-known string analogues.
Is there a regular tree language in which the average height of a tree of size $n$ is neither $\Theta(n)$ nor $\Theta(\sqrt{n})$?
Obviously there are tree languages such that the height of a tree is linear in its size; and in the book Analytic Combinatorics it is shown e.g. that binary trees of size $n$ have average height $2\sqrt{ \pi n}$. If I understand Proposition VII.16 (p.537) of the mentioned book correctly, then there is a wide subset of regular tree languages that have average height of $\Theta(\sqrt{n})$, namely those in which the tree language is also a simple variety of trees fulfilling some extra conditions.
So I was wondering whether there is a regular tree language showing a different average height or if there is a true dichotomy for regular tree languages.
