  We study the equivalence classes under $\Delta^1_1$ isomorphism, otherwise
effective-Borel isomorphism, between complete separable metric spaces which
admit a recursive presentation and we show the existence of strictly increasing
and strictly decreasing sequences as well as of infinite antichains under the
natural notion of $\Delta^1_1$-reduction, as opposed to the non-effective case,
where only two such classes exist, the one of the Baire space and the one of
the naturals. A key tool for our study is a mapping $T \mapsto \mathcal{N}^T$
from the space of all trees on the naturals to the class of Polish spaces, for
which every recursively presented space is $\Delta^1_1$-isomorphic to some
$\mathcal{N}^T$ for a recursive $T$, so that the preceding spaces are
representatives for the classes of $\Delta^1_1$-isomorphism. We isolate two
large categories of spaces of the type $\mathcal{N}^T$, the Kleene spaces and
the Spector-Gandy spaces and we study them extensively. Moreover we give
results about hyperdegrees in the latter spaces and characterizations of the
Baire space up to $\Delta^1_1$-isomorphism.
