09-07-2019, 07:02 AM
And rotations come in four flavors depending on how the imbalance sits along that path. You look at the child and grandchild to pick which turn to make. I usually spot a left left case when both the parent and the new node lean the same way. Then a single right rotation brings everything back into line without much fuss. Or maybe you run into a left right mess where the child leans opposite the parent. You perform a left rotation on the child first and follow it with a right rotation on the parent to straighten the whole mess. But you always update heights after each spin so future checks stay accurate.
Now the height of a node equals one plus the bigger height from its two kids. I calculate that every time a node gets touched during the climb back to the root. You avoid letting any balance factor stray beyond one or minus one because that keeps the tree from turning into a slow chain. And the whole climb takes log time on average since the tree stays short. Perhaps you wonder why we bother with all this extra checking instead of just adding nodes blindly. I tell you the extra work pays off later when searches stay quick even after thousands of inserts. But you still have to watch for the four imbalance patterns each time.
You trace the path again in your mind when a right right case appears. I rotate left once on the parent and the tree levels out nicely. Or the right left pattern shows up and you rotate right on the child then left on the parent to settle it. And each rotation preserves the search order so nothing gets lost or duplicated. I like how the method keeps everything local so you only touch a few nodes per insert. Perhaps the tree grows lopsided on one side and you catch it early before it spreads. You update every height on the way up because skipping that step leads to wrong balance calls later.
But sometimes the imbalance sits far up the path and you might need only one rotation to fix multiple spots at once. I have seen cases where a single spin corrects the whole chain below it. You practice spotting the exact pattern by drawing the nodes on paper first. And the log n bound stays solid because rotations never make the tree taller than it needs to be. Perhaps you insert a batch of numbers in order and watch the rotations flip things around to stay flat. I always recompute the factor right after the spin so the next ancestor check works correctly.
You notice the process repeats until you reach the root or find a node whose balance stays fine. And stopping early saves a little work when lower fixes already restored order. I find the method elegant once you get the hang of reading the lean directions. But you still run the height math on every insert to keep things honest. Perhaps the tree holds strings or other data instead of numbers and the same rules apply without change. You compare the keys the same way and rotate based on structure alone. And the end result stays balanced no matter what order the inserts arrive in.
You keep the conversation going by trying small examples yourself and tracing each step. I think that hands on feel helps lock in when to pick which rotation. But the core idea stays simple even if the cases look tricky at first glance. And you end up with a tree that never stretches more than log n levels deep. Perhaps later you add delete logic and the same rotations pop up again in slightly different spots. I have walked through both operations many times and they share the same height tracking habit.
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