Yes. Even better, a good database will avoid traversing the blocks and path, when the key is close enough to the previous lookup's hint, making all but the fir

Yes. Even better, a good database will avoid traversing the blocks and path, when the key is close enough to the previous lookup's hint, making all but the first lookup in a nearly-consecutive sequence take O(1) instead of O(log N) where N is the size of the tree, and runs of n nearly-consecutive operations take O(n + log N) instead of O(n log N).