Hi, I apologize if this question is poorly formed as I'm not a graph theorist. I've figured out how to encode some information about a problem in surface topology in the language of graph theory. I just can't figure out how to prove it in graph theory or where to look for more information.

Given some $n$, suppose I have two sets of vertices $V=\{v_1,...,v_k\}$ and $W=\{w_1,...,w_l\}$ with $k,l$ strictly less than $n$ where each vertex is assigned a degree $d_{v_i}$ or $d_{w_j}$, respectively. Suppose I know there exists a bipartite graph with $n$ edges from $V$ to $W$ respecting the given degrees of each vertex that is a forest. I would like to be able to say that this implies every bipartite graph with $n$ edges from $V$ to $W$ that respects the given degrees contains a component that is a tree. Is there any chance this is true or any references that could help me figure it out? I do also have the additional information that the only cycles I can allow are 2-cycles. Does this help anything?