Packing large balanced trees into bipartite graphs

We prove that for every ${\gamma > 0}$ there exists $n_0 \in \mathbb{N}$ such that for every ${n \geq n_0}$ any family of up to $\lfloor{n^{\frac12+\gamma}}\rfloor$ trees having at most $(1-\gamma)n$ vertices in each bipartition class can be packed into $K_{n,n}$. As a tool for our proof, we show an approximate bipartite version of the Koml\'os-S\'ark\"ozy-Szemer\'edi Theorem, which we believe to be of independent interest.