Regular Leaf Languages and (non-) Regular Tree Shapes

We investigate the power of regular leaf languages with respect to three diierent tree shapes: the rst one is the case of arbitrary (and thus potentially non-regular) computation trees; the second one is the case of balanced computation trees (these trees are initial segments of full binary trees); and the third one is the case of full binary computation trees. It is known that the class characterizable by regular leaf languages is PSPACE in all cases; but, contrary to our intuition, the description power does not decrease, but may increase with the regularity of the trees. Especially we can show that all complexity classes charac-terizable by regular leaf languages and arbitrary computation trees are also characterizable by regular leaf languages and balanced computation trees; and, similarly, all complexity classes characterizable by regular leaf languages and balanced computation trees are also characterizable by regular leaf languages and full binary computation trees. It remains unknown, whether the respective converse statements hold, too. We also investigate three subclasses of the class of regular languages and their description power with respect to the three diierent models. We exhibit the somehow strange fact that for each of the three models there is one subclass, which makes this model in question strictly more powerful than the other ones (under some reasonable complexity theoretic assumption). We conclude that the crucial property behind the characterization power is not a property of the model, but a matter of the extent, to which the models and the respective leaf language classes are suitable for each other.