  A recent result of Condon, Kim, K\"{u}hn and Osthus implies that for any
$r\geq (\frac{1}{2}+o(1))n$, an $n$-vertex almost $r$-regular graph $G$ has an
approximate decomposition into any collections of $n$-vertex bounded degree
trees. In this paper, we prove that a similar result holds for an almost
$\alpha n$-regular graph $G$ with any $\alpha>0$ and a collection of bounded
degree trees on at most $(1-o(1))n$ vertices if $G$ does not contain large
bipartite holes. This result is sharp in the sense that it is necessary to
exclude large bipartite holes and we cannot hope for an approximate
decomposition into $n$-vertex trees.
  Moreover, this implies that for any $\alpha>0$ and an $n$-vertex almost
$\alpha n$-regular graph $G$, with high probability, the randomly perturbed
graph $G\cup \mathbf{G}(n,O(\frac{1}{n}))$ has an approximate decomposition
into all collections of bounded degree trees of size at most $(1-o(1))n$
simultaneously. This is the first result considering an approximate
decomposition problem in the context of Ramsey-Tur\'an theory and the randomly
perturbed graph model.
