  In this paper, we present a new method to produce explicit formulas for the
number of rooted spanning forests $f(n)$ for the infinite family of graphs
$H_n=H_n(G_1,\,G_2,\ldots,G_m)$ obtained as a circulant foliation over a graph
$H$ on $m$ vertices with fibers $G_1,\,G_2,\ldots,G_m.$ Each fiber
$G_i=C_n(s_{i,1},\,s_{i,2},\ldots,s_{i,k_i})$ of this foliation is the
circulant graph on $n$ vertices with jumps
$s_{i,1},\,s_{i,2},\ldots,s_{i,k_i}.$ This family includes the family of
generalized Petersen graphs, $I$-graphs, sandwiches of circulant graphs,
discrete torus graphs and others.
  The formulas are expressed through Chebyshev polynomials. We prove that the
number of rooted spanning forests can be represented in the form
$f(n)=p\,f(H)a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a
prescribed natural number depending on the number of odd elements in the set of
$s_{i,j}.$ Finally, we find an asymptotic formula for $f(n)$ through the Mahler
measure of the associated Laurent polynomial.
