Iterative version of Binary Tree Search

Question

Basic Tree-search algorithm for searching a node (with value k) in a binary search tree. 'x' denotes the node of the binary search tree.

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

Iterative Version:

ITERATIVE-TREE-SEARCH(x, k)
 while x ≠ NIL and k ≠ key[x]
     do if k < key[x]
           then x ← left[x]
           else x ← right[x]
 return x

Shouldn't the first line (of the iterative algorithm) actually be while (x ≠ NIL OR k ≠ key[x]) instead of (while x ≠ NIL and k ≠ key[x]) ?

By the way, if you were wondering, this is from one of the famous books of Algorithm Analysis.

Answer 1

No, it needs to be and because otherwise you'll dereference NIL if k isn't found. Remember that while executes as long as the expression evaluates to true.

while x ≠ NIL and k ≠ key[x]

If x is NIL, then the expression x ≠ NIL and k ≠ key[x] is false, because x ≠ NIL is false. Either side of an and being false makes the whole expression false.

If or were used instead, x ≠ NIL would still be false, but you'd need to evaluate the other side — both sides of an or must be false for the or to be false. Unfortunately, evaluating the other side dereferences NIL. Ooops. Even if it weren't for that problem, k ≠ key[x] is true (because we're considering the case where k isn't found, so no key in the tree x can be k). Since one (or more) sides of the or is true, the or evaluates to true, and the loop continues forever.

In English, that while can read: while there is still tree left (x ≠ NIL) and we haven't yet found what we're looking for (k ≠ key[x]).