In this paper we present new algorithms for counting minimum cost bounded degree subtrees in connected graphs in which the 2-vertex-connected (biconnected) components have small sizes. The 2-vertex-connected components and the cut vertices can be organized into a block-cut vertex tree which is also a tree decomposition with small width of the graph. We present a dynamic programming algorithm which is very efficient on this particular tree decomposition and we also discuss methods of solving the

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... problem given an arbitrary tree decomposition with small width. Among some of the most important results is an algorithm which can efficiently compute the number of subtrees of a (small) graph corresponding to each possible degree sequence.