  We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely
intersect every hyperplane absolutely winning (HAW) set with full Hausdorff
dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realization of a
fractal percolation process, then almost surely (conditioned on
$E\neq\emptyset$), for every countable collection
$\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of
$\mathbb{R}^{d}$,
$\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$,
where $\text{BA}_{d}$ is the set of badly approximable vectors in
$\mathbb{R}^{d}$. We show this by proving that $E$ almost surely contains
hyperplane diffuse subsets which are Ahlfors-regular with dimensions
arbitrarily close to $\dim_{H}\left(E\right)$.
  We achieve this by analyzing Galton-Watson trees and showing that they almost
surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield
the aforementioned subsets of $E$. This method allows us to obtain a more
general result by projecting the Galton-Watson trees against any similarity IFS
whose attractor is not contained in a single affine hyperplane. Thus our
general result relates to a broader class of random fractals than fractal
percolation.
