Is perfect matching for bipartite graph with no cycles unique?
If the graph is acyclic, which implies that it is also bipartite, then the perfect matching is unique by the following algorithm: 

While the graph is not empty, pick a leaf vertex $u$ (which exists because the graph is acyclic) add the edge between $u$ and its unique neighbor $v$ to the matching and then remove $u$ and $v$. If at some point graphs without incident edges are produced, then there is no perfect matching. 

This is probably folklore knowledge which would mean that there is no citable source.