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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 anapproximate decomposition into any collections of $n$-vertex bounded degreetrees. 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 boundeddegree trees on at most $(1-o(1))n$ vertices if $G$ does not contain largebipartite holes. This result is sharp in the sense that it is necessary toexclude large bipartite holes and we cannot hope for an approximatedecomposition 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 perturbedgraph $G\cup \mathbf{G}(n,O(\frac{1}{n}))$ has an approximate decompositioninto all collections of bounded degree trees of size at most $(1-o(1))n$simultaneously. This is the first result considering an approximatedecomposition problem in the context of Ramsey-Tur\'an theory and the randomlyperturbed graph model.

Tree decompositions of graphs without large bipartite holes