Let $G$ be a bipartite graph on $n$ vertices of either color.
Suppose $G$ contains no perfect matching the number of edges can be $Omega(n^2)$ (just do not place an edge between a particular pair of vertices).
Suppose $G$ contains exactly one perfect matching what is the maximum number of edges it can have?
Suppose $G$ contains exactly two perfect matchings what is the maximum number of edges it can have?
I think $2n-1$ is the upper bound for 1. Essentially fix a perfect matching. As long as we do not have a $2times2$ subgraph we can add edges and there is exactly $n-1$ additional edges we can add. For 2. the upper limit is $2n$ I think by the same calculation.