  For any Polish space $X$ it is well-known that the Cantor-Bendixson rank
provides a co-analytic rank on $F_{\aleph_0}(X)$ if and only if $X$ is a
$\sigma$-compact. In the case of $\omega^\omega$ one may recover a co-analytic
rank on $F_{\aleph_0}(\omega^\omega)$ by considering the Cantor-Bendixson rank
of the induced trees instead. In this paper we will generalize this idea to
arbitrary Polish spaces and thereby construct a family of co-analytic ranks on
$F_{\aleph_0}(X)$ for any Polish space $X$. We study the behaviour of this
family and compare the ranks to the original Cantor-Bendixson rank. The main
results are characterizations of the compact and $\sigma$-compact Polish spaces
in terms of this behaviour.
