  The notion of $\mathcal{H}$-treewidth, where $\mathcal{H}$ is a hereditary
graph class, was recently introduced as a generalization of the treewidth of an
undirected graph. Roughly speaking, a graph of $\mathcal{H}$-treewidth at most
$k$ can be decomposed into (arbitrarily large) $\mathcal{H}$-subgraphs which
interact only through vertex sets of size $O(k)$ which can be organized in a
tree-like fashion. $\mathcal{H}$-treewidth can be used as a hybrid
parameterization to develop fixed-parameter tractable algorithms for
$\mathcal{H}$-deletion problems, which ask to find a minimum vertex set whose
removal from a given graph $G$ turns it into a member of $\mathcal{H}$. The
bottleneck in the current parameterized algorithms lies in the computation of
suitable tree $\mathcal{H}$-decompositions.
  We present FPT approximation algorithms to compute tree
$\mathcal{H}$-decompositions for hereditary and union-closed graph classes
$\mathcal{H}$. Given a graph of $\mathcal{H}$-treewidth $k$, we can compute a
5-approximate tree $\mathcal{H}$-decomposition in time $f(O(k)) \cdot n^{O(1)}$
whenever $\mathcal{H}$-deletion parameterized by solution size can be solved in
time $f(k) \cdot n^{O(1)}$ for some function $f(k) \geq 2^k$. The current-best
algorithms either achieve an approximation factor of $k^{O(1)}$ or construct
optimal decompositions while suffering from non-uniformity with unknown
parameter dependence. Using these decompositions, we obtain algorithms solving
Odd Cycle Transversal in time $2^{O(k)} \cdot n^{O(1)}$ parameterized by
$\mathsf{bipartite}$-treewidth and Vertex Planarization in time $2^{O(k \log
k)} \cdot n^{O(1)}$ parameterized by $\mathsf{planar}$-treewidth, showing that
these can be as fast as the solution-size parameterizations and giving the
first ETH-tight algorithms for parameterizations by hybrid width measures.
