Packing two copies of a tree into a bipartite graph with restrained maximum degree

For a bipartite graph $G(U_1,U_2)$, we use ${\Delta }_1\left(G\right)$ and ${\Delta }_2\left(G\right)$ to denote the maximal degree of the vertices in the sets $U_1$ and $U_2$ in $G$, respectively. Let the tree $T(V_1,V_2)$ be an $(a,b)$-bipartite graph of order $n$. We prove that if $|a-b|\le 2$, then there is a 2-packing $(\sigma ,\tau )$ of $T$ in some complete bipartite graph $B_{n+1}(X_1,X_2)$ such that for each $i\in \{1,2\}$, ${\Delta }_i(\sigma \left(T\right)\cup \tau \left(T\right))$ is at most ${\Delta }_i\left(T\right)+2$, where $V_i\subseteq X_i(i=1,2)$. This result is sharp because ${\Delta }_i\left(T\right)+2$ cannot be reduced any further. By applying this result, we deduce a corollary for the existence of packing of three bipartite graphs into complete bipartite graphs.