Why Depth is O(log n)

When you remove a centroid, each component has at most $n/2$ nodes. At the next level, components have at most $n/4$ nodes. Then $n/8$, and so on. Each level halves the maximum component size. After $k$ levels, components have at most $n / 2^k$ nodes.

When this reaches $1$, you are done. Solving $n / 2^k = 1$ gives $k = \log_2 n$, so the depth is logarithmic. So the decomposition tree has $O(\log n)$ depth, which bounds the recursion depth. This is why centroid decomposition is efficient.

Space complexity is $O(n)$ for the data structures used.