Given a binary search tree (BST), find the lowest common ancestor of two given nodes in the BST.
_______6______ / \ ___2__ ___8__ / \ / \ 0 _4 7 9 / \ 3 5
Using the above tree as an example, the lowest common ancestor (LCA) of nodes 2 and 8 is 6. But how about LCA of nodes 2 and 4? Should it be 6 or 2?
According to the definition of LCA on Wikipedia: “The lowest common ancestor is defined between two nodes vand w as the lowest node in T that has both v and w as descendants (where we allow a node to be a descendant of itself).” Since a node can be a descendant of itself, the LCA of 2 and 4 should be 2, according to this definition.
Hint:
A top-down walk from the root of the tree is enough.
Solution:
There’s only three cases you need to consider.
- Both nodes are to the left of the tree.
- Both nodes are to the right of the tree.
- One node is on the left while the other is on the right.
For case 1), the LCA must be in its left subtree. Similar with case 2), LCA must be in its right subtree. For case 3), the current node must be the LCA.
Therefore, using a top-down approach, we traverse to the left/right subtree depends on the case of 1) or 2), until we reach case 3), which we concludes that we have found the LCA.
A careful reader might notice that we forgot to handle one extra case. What if the node itself is equal to one of the two nodes? This is the exact case from our earlier example! (The LCA of 2 and 4 should be 2). Therefore, we add case number 4):
The run time complexity is O(h), where h is the height of the BST. Translating this idea to code is straightforward (Note that we handle case 3) and 4) in the else statement to save us some extra typing ):
Node *LCA(Node *root, Node *p, Node *q) {
if (!root || !p || !q) return NULL;
if (max(p->data, q->data) < root->data)
return LCA(root->left, p, q);
else if (min(p->data, q->data) > root->data)
return LCA(root->right, p, q);
else
return root;
}
From:
http://leetcode.com/2011/07/lowest-common-ancestor-of-a-binary-search-tree.html