  The third author noticed in his 1992 PhD Thesis [Sim92] that every regular
tree language of infinite trees is in a class $\Game (D\_n({\bf\Sigma}^0\_2))$
for some natural number $n\geq 1$, where $\Game$ is the game quantifier. We
first give a detailed exposition of this result. Next, using an embedding of
the Wadge hierarchy of non self-dual Borel subsets of the Cantor space
$2^\omega$ into the class ${\bf\Delta}^1\_2$, and the notions of Wadge degree
and Veblen function, we argue that this upper bound on the topological
complexity of regular tree languages is much better than the usual
${\bf\Delta}^1\_2$.
