TSP in bounded tree-width (or bounded branch width) graphs
TSP is the optimization version of HAMILTONIAN CYCLE, which can be expressed by an MSO-formula (Monadic Second-Order Logic) with edge-set quantifications ($MSO_2$, also known as MSO in the incidence-graph model) as follows:  Let $hamilton(F)$ express that $F$ is a set of edges of the input graph $G = (V, E)$ such that every vertex is incident to exactly two edges in $F$, and additionally $F$ is connected.   Then
$$\min_{F \subseteq E} \{ \sum_{e \in F} w(e) : G \models hamilton(F)\},$$
where $w(e)$ denodes the weight of an edge $e$.  Hence, TSP is a Linear MSO-definable Optimization problem.   Via Courcelle's Theorem, TSP can therefore be solved  in linear time on graphs of bounded treewidth (for example, using dynamic programming as mentioned by Andrew in the comment above).