  Let $N\geq 2$ be an integer, a (1, $N$)-periodic graph $G$ is a periodic
graph whose vertices can be partitioned into two sets
$V_1=\{v\mid\sigma(v)=v\}$ and $V_2=\{v\mid\sigma^i(v)\neq v\ \mbox{for any}\
1<i<N\}$, where $\sigma$ is an automorphism with order $N$ of $G$. The subgraph
of $G$ induced by $V_1$ is called a fixed subgraph. Yan and Zhang [Enumeration
of spanning trees of graphs with rotational symmetry, J. Comb. Theory Ser. A,
118(2011): 1270-1290] studied the enumeration of spanning trees of a special
type of (1, $N$)-periodic graphs with $V_1=\emptyset$ for any non-trivial
automorphism with order $N$. In this paper, we obtain a concise formula for the
number of spanning trees of (1, $N$)-periodic graphs. Our result can reduce to
Yan and Zhang's when $V_1$ is empty. As applications, we give a new closed
formula for the spanning tree generating function of cobweb lattices, and
obtain formulae for the number of spanning trees of circulant graphs
$C_n(s_1,s_2,\ldots,s_k)$ and $K_2\bigvee C_n(s_1,s_2,\ldots,s_k)$.
