在修改后的二进制搜索树上搜索，插入和删除

Question

Say that each node in a binary search tree x keeps x.successor instead of x.parent. Describe Search, Insert, and Delete algorithms using pseudo code that operate in O(h) where h is the height of the tree. So far, I believe that the simple binary search tree algorithm for search still applies.

TREE-SEARCH(x, k)
if x == NIL or k == x.key 
    return x
if k < x.key
    return TREE-SEARCH(x.left, k)
else return TREE-SEARCH(x.right, k)

This algorithm should be unaffected by the modification and clearly runs in O(h) time. However, for the Insert and Delete algorithms the modification changes the straightforward way to do these. For example, here is the algorithm for Insertion using a normal binary search tree, T.

TREE-INSERT(T,z)
y = NIL
x = T.root
while x =/= NIL
    y = x 
    if z.key < x.key
        x = x.left
    else x = x.right
z.p = y
if y == NIL
    T.root = z    //Tree T was empty
elseif z.key < y.key
    y.left = z
else y.right = z

Clearly we cannot use zp because of our modification to the binary search tree. It has been suggested that a subroutine that finds the parent be implemented, but I fail to see how this can be done.

Answer 1

Searching will be unaffected

Sketch of an answer

While inserting, we want the successor of the newly inserted node. If you know the algorithm for the successor in a normal binary search tree, you'd know that at the leaf node, the successor is the first ancestor whose left child is an ancestor of the node. So while doing normal insert the last node whose left child we follow will be the successor of the node. On this insert path the last node where you follow the right child will be the one whose successor is our current node.

For deletion, if in x.successor you are storing the pointer to the successor node, it becomes simple, just replace the node-to-be-deleted's key with the successor's key and the node.successor pointer with the successors pointer. If you're just storing the key, you need to update the node that had this node as the successor. the candidates are, if it has a left child, the rightmost node in this subtree. If it doesn't, search for the node and the last node where you move to the right child will be the required node.

I don't know how to draw diagrams here, it's simpler with the diagrams.