TSP in bounded tree-width (or bounded branch width) graphs I see there are some papers/thesis which says TSP is solvable in $O(n)$ in bounded tree-width graphs and some of them implicitly refer to Cook and Seymour (not exact paper just said that they solve it), but I can't find any paper about it.

All the things I can find is branch-decomposition based algorithms with merging techniques, which are all heuristic (e.g very nice heuristic) but non of them are exact algorithm. Does anyone know the related papers for linear time solution? Or paper which says is impossible? or probably impossible? (any good news bad news, its not 3in1 question:).
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).