  The Erd\H{o}s-S\'{o}s Conjecture states that every graph with average degree
more than $k-2$ contains all trees of order $k$ as subgraphs. In this paper, we
consider a variation of the above conjecture: studying the maximum size of an
$(n,m)$-bipartite graph which does not contain all $(k,l)$-bipartite trees for
given integers $n\ge m$ and $k\ge l$. In particular, we determine that the
maximum size of an $(n,m)$-bipartite graph which does not contain all
$(n,m)$-bipartite trees as subgraphs (or all $(k,2)$-bipartite trees as
subgraphs, respectively). Furthermore, all these extremal graphs are
characterized.
