Size of tree decomposition Given a graph $G$ with $n$ vertices, let $(X, T)$ be a tree decomposition of $G$ with the smallest width. Is the number of nodes in $T$ upper bounded by $n$? I have googled it but all materials I found discuss only treewidth.
Not a complete answer, but in T. Kloks Treewidth you have the following lemma:


  There always exists a nice tree decomposition of width $k$ and with at most $4n$ bags.


It is NP-complete to determine the minimum number of bags in a tree decomposition of width $k$.