When you remove a centroid, each component has at most nodes. At the next level, components have at most nodes. Then , and so on. Each level halves the maximum component size. After levels, components have at most nodes.
When this reaches , you are done. Solving gives , so the depth is logarithmic. So the decomposition tree has depth, which bounds the recursion depth. This is why centroid decomposition is efficient.
Space complexity is for the data structures used.