This paper investigates the possibility to find a single FPT algorithm with respect to the treewidth that solves a large variety of spanning tree, steiner tree and more generally covering tree problems that can be found in the literature. This includes problems for which no such algorithm was already described as the Minimum Branch Vertices problem, the Minimum Leaf Spanning Tree problem or the k-Bottleneck Steiner Tree Problem. To do so, a generalization of many of those covering tree problems, called the Minimum subTree problem with Degree Weights MTDW, is introduced and the parameterized complexity of that problem is studied.