  Let $G$ be a graph and let Pm$(G)$ denote the number of perfect matchings of
$G$. We denote the path with $m$ vertices by $P_m$ and the Cartesian product of
graphs $G$ and $H$ by $G\times H$. In this paper, as the continuance of our
paper [19], we enumerate perfect matchings in a type of Cartesian products of
graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some
of our results: 1. Let $T$ be a tree and let $C_n$ denote the cycle with $n$
vertices. Then Pm$(C_4\times T)=\prod (2+\alpha^2)$, where the product ranges
over all eigenvalues $\alpha$ of $T$. Moreover, we prove that Pm$(C_4\times T)$
is always a square or double a square. 2. Let $T$ be a tree. Then Pm$(P_4\times
T)=\prod (1+3\alpha^2+\alpha^4)$, where the product ranges over all
non-negative eigenvalues $\alpha$ of $T$. 3. Let $T$ be a tree with a perfect
matching. Then Pm$(P_3\times T)=\prod (2+\alpha^2),$ where the product ranges
over all positive eigenvalues $\alpha$ of $T$. Moreover, we prove that
Pm$(C_4\times T)=[{Pm}(P_3\times T)]^2$.
