Point update on node : update position tin[v] in the flattened array. This is a single-point update in your segment tree or BIT. Range update on subtree: update range [tin[v], tout[v]].
If your segment tree supports lazy propagation, this is . Path updates (root to ) with entry-exit trick: update positions tin[v] and tout[v] with add/subtract values.
Space complexity is for the data structures used.