Fix a parameter D and let the $D$-treewidth be the minimum width among tree-decompositions of H where the underlying tree has height D.

We know:

- 1-tw $= |V(H)|$.
- 2-tw $\leq vc(H)$, but this is not tight, as the matching has small 2-tw. Here $vc(H)$ is vertex cover number.
- b-tw $\leq 4 \cdot tw(H) + 3$ for $b \geq O(log |V(H)|)$, by this paper: https://research-explorer.ista.ac.at/download/5427/5471/IST-2014-314-v1%2B1_long.pdf

Can we say something in general about D-treewidth? In particular, how does $D$-treewidth decrease as we increase $D$.