  We prove that if $N\ge 2$ and $\alpha: F_N\to \pi_1(\Gamma)$ is a marking on
$F_N$, then for any integer $r\ge 2$ and any $F_N$-invariant collection of
non-negative integral "weights" associated to all subtrees $K$ of $\widetilde
\Gamma$ of radius $\le r$ satisfying some natural "switch" conditions, there
exists a finite cyclically reduced folded $\Gamma$-graph $\Delta$ realizing
these weights as numbers of "occurrences" of $K$ in $\Delta$. As an
application, we give a new, more direct and explicit, proof of one of the main
results of our paper with Nagnibeda \cite{KN3} stating that for any $N\ge 2$
the set $\gcnr$ of all rational subset currents is dense in the space $\gcn$ of
subset currents on $F_N$. We also answer one of the questions (Problem 10.11)
posed in \cite{KN3}. Thus we prove that if a nonzero $\mu\in \gcn$ has all
weights with respect to some marking being integers, then $\mu$ is the sum of
finitely many "counting" currents corresponding to nontrivial finitely
generated subgroups of $F_N$.
