Evaluating the Tutte Polynomial for Graphs of Bounded Tree-Width

It is known that evaluating the Tutte polynomial, T(G; x, y), of a graph, G, is nP-hard at all but eight specific points and one specific curve of the (x, y)-plane. In contrast we show that if k is a fixed constant then for graphs of tree-width at most k there is an algorithm that will evaluate the polynomial at any point using only a linear number of multiplications and additions.