A tree $G$ on $n$ vertices $V=\{v\_1,...,v\_n\}$ is a connected undirected graph which is acyclic. For each tree $G$ one can split the set of vertices $V$ into two disjoint subsets $U,W \subset V$ such that $V = U \cup W$ and each edge in $G$ is between a vertex from $U$ and a vertex from $W$. In order to get uniqueness of $U$ and $W$ let's require $v\_1$ to be in $U$.


My question is if there is a known formula for the number of trees on $n$ vertices which satisfy $\#U = n\_1$ and $\#W = n\_2$ for given $n\_1,n\_2$ with $n\_1+n\_2=n$.


Any help is much appreciated.



Is there a formula for the number of trees with this extra condition?
If the trees are labelled, then each tree satisfying the condition corresponds to exactly one spanning tree of the bipartite graph $K\_{n\_1,n\_2}$. Therefore the answer is the number of spanning trees of the bipartite graph $K\_{n\_1,n\_2}$, which is $n\_1^{n\_2-1}n\_2^{n\_1-1}$ according to [this](https://math.stackexchange.com/questions/3781460/spanning-trees-of-the-complete-bipartite-graph).


