Tree decompositions of optimal width where every vertex is in a bounded number of bags? Let $G$ be a graph on $n$ vertices whose maximum degree is at most $\Delta$ and whose treewidth is at most $k$.
Does there exist a function $f(k, \Delta)$, independent of $n$, such that it is possible to find a tree decomposition of $G$ of width $k$ (i.e. with bags of size $k+1$) with the additional property that every vertex of $G$ is in at most $f(k, \Delta)$ of the bags?

A related concept which is perhaps relevant is that of domino treewidth.
We say a tree decomposition is domino if every vertex is in at most $2$ of the bags.
Bodlaender and Engelfriet [ https://link.springer.com/chapter/10.1007/3-540-59071-4_33 ] show that there exists $c(k, \Delta)$ such that every graph with maximum degree at most $\Delta$ and treewidth at most $k$ has a domino tree decomposition of width $c(k, \Delta)$.

Thus, by increasing the size of the bags from $k$ to $c(k, \Delta)$, they can find a new tree decomposition where each vertex is in at most $2$ bags (and clearly the $2$ cannot be lowered).
It seems to me that there is a trade-off between "size of the bags" and "number of bags per vertex" for the class of graphs with bounded degree and treewidth, and my question is if it possible to control the number of "bags per vertex" if we insist on not increasing the bag size.
Answering my own question: the answer is "no". For each $k \ge 3$, Ding and Oporowski construct a graph $G_k$ on $2k$ vertices with treewidth $3$, such that in every optimal tree decomposition of $G_k$ there is a vertex which is in at least $2k-2$ bags (see Figure 1 in their paper).

Ding, Guoli; Oporowski, Bogdan, Some results on tree decomposition of graphs, J. Graph Theory 20, No. 4, 481-499 (1995). ZBL0837.05044.