Fair deletion of nodes in Binary Search Tree

Question

The idea of deleting a node in BST is:

1. If the node has no child, delete it and update the parent's pointer to this node as null

2. If the node has one child, replace the node with its children by updating the node's parent's pointer to its child

3. If the node has two children, find the predecessor of the node and replace it with its predecessor, also update the predecessor's parent's pointer by pointing it to its only child (which only can be a left child)

the last case can also be done with use of a successor instead of predecessor!

It's said that if we use predecessor in some cases and successor in some other cases (giving them equal priority) we can have better empirical performance ,

Now the question is , how is it done ? based on what strategy? and how does it affect the performance ? (I guess by performance they mean time complexity)

What I think is that we have to choose predecessor or successor to have a more balanced tree ! but I don't know how to choose which one to use !

One solution is to randomly choose one of them (fair randomness) but isn't better to have the strategy based on the tree structure ? but the question is WHEN to choose WHICH ?

Answer 1

As you said, it's a question of balance, so in general the method that disturbs the balance the least is preferable. You can hold some metrics to measure the level of balance (eg, difference from maximal and minimal leaf height, average height etc.), but I'm not sure whether the overhead worth it. Also, there are self-balancing data structures (red-black, AVL trees etc.) that mitigate this problem by rebalancing after each deletion. If you want to use the basic BST, I suppose the best strategy without apriori knowledge of tree structure and the deletion sequence would be to toggle between the 2 methods for each deletion.

Answer 2

The thing is that is fundamental problem - to find correct removal algorithm for BST. For 50 years people were trying to solve it (just like in-place merge) and they didn't find anything better then just usual algorithm (with predecessor/successor removing). So, what is wrong with classic algorithm? Actually, this removing unbalances the tree. After several random operations add/remove you'll get unbalanced tree with height sqrt(n) . And it is no matter what you choosed - remove successor or predecessor (or random chose beetwen these ways) - the result is the same.

So, what to choose? I'm guessing random based (succ or pred) deletion will postpone unbalancing of your tree. But, if you want to have perfectly balanced tree - you have to use red-black ones or something like that.