Lowest Common Ancestor of a Binary Search Tree (BST)

July 17, 2011 in binary tree

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 v and 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.

  1. Both nodes are to the left of the tree.
  2. Both nodes are to the right of the tree.
  3. 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):

4. When the current node equals to either of the two nodes, this node must be the LCA too.

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 ;) ):

You should realize quickly that the above code contains tail recursion. Converting the above function to an iterative one is left as an exercise to the reader.

Further Thoughts:
What if you are required to find the LCA of a binary tree (not a BST)? Check out my next post: Lowest Common Ancestor of a Binary Tree Part I to find out how.

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33 responses to Lowest Common Ancestor of a Binary Search Tree (BST)

  1. I think you are missing the case when either p or q is not a member of given tree.
    Example: In your tree above, find LCA of 1 and 4. In that case your code would return 2, even when 1 is not present in tree.

    VA:F [1.9.22_1171]
    -8
    • I agree with you. And plus the case that one of them is not in the tree. But I question how far we should go. What if the given tree is not a valid tree? Do we need to take care of that? I think we should just ask the interviewers to clarify and move on from there.

      VN:F [1.9.22_1171]
      +6
      • We won’t have to, the prerequisite is the tree is a valid binary search tree

        VA:F [1.9.22_1171]
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  2. We need to check that both the nodes are present in tree (log N complexity), and then find LCA.

    If the query is not valid, we should return null. Given below is wrapper.

    VA:F [1.9.22_1171]
    +3
  3. I think your soln. doesn’t work for finding LCA of 4 and 5 – which should be 2 but the code mention returns 4, which is incorrect.

    VA:F [1.9.22_1171]
    -1
    • You can easily fix it by adding more if conditions,

      if (root == null || root.data == a.data || root.data == b.data)
      return null;

      if (root.RightChild != null && (root.RightChild.data == a.data || root.RightChild.data == b.data))
      return root;

      if (root.LeftChild != null && (root.LeftChild.data == a.data || root.LeftChild.data == b.data))
      return root;

      VA:F [1.9.22_1171]
      0
    • No, the code is correct, it should be 4 only.
      Quoting from wikipedia: “The lowest common ancestor (LCA) is a concept in graph theory and computer science. Let T be a rooted tree with n nodes. The lowest common ancestor is defined between two nodes v and 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).”

      Refer here – http://en.wikipedia.org/wiki/Lowest_common_ancestor

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    • The correct answer is 4.

      VA:F [1.9.22_1171]
      0
  4. It seems that the problem shall be clarified in the way that:
    Not 2 nodes in the binary search tree can have the same value.

    Simply specifying that a BST would not eleminate the above situation since:
    binarysearch-tree property:
    • Let x be a node in a binary search tree. If y is a node in the left subtree of x, then key[y] ≤ key[x]. If y is a node in the right subtree of x, then key[x] ≤ key[y].

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  5. I have a question:
    Is doing
    Is

    worse/wrong than using the max

    that you use now?

    VA:F [1.9.22_1171]
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  6. I mean if (p->data data < data)

    VA:F [1.9.22_1171]
    0
  7. #include
    using namespace std;

    struct Node
    {
    Node(int value) : data(value)
    {
    pLeft = NULL;
    pRight = NULL;
    }
    int data;
    Node* pLeft;
    Node* pRight;
    };

    Node* LowestCommonParent(Node* pBSTRoot, Node* pNode1, Node* pNode2)
    {
    if(NULL == pBSTRoot || NULL == pNode1 || NULL == pNode2)
    return NULL;
    Node* pLess = NULL;
    Node* pGreater = NULL;
    if(pNode1->data data)
    {
    pLess = pNode1;
    pGreater = pNode2;
    }
    else if(pNode1->data > pNode2->data)
    {
    pLess = pNode2;
    pGreater = pNode1;
    }
    else
    {
    return NULL;
    }
    if((pLess->data data && pGreater->data > pBSTRoot->data) ||
    (pLess->data data && pGreater->data >= pBSTRoot->data))
    {
    return pBSTRoot;
    }
    else if(pGreater->data data)
    {
    return LowestCommonParent(pBSTRoot->pLeft, pLess, pGreater);
    }
    else if(pLess->data > pBSTRoot->data)
    {
    return LowestCommonParent(pBSTRoot->pRight, pLess, pGreater);
    }
    }
    int main()
    {
    Node node0(6);
    Node node1(2);
    Node node2(8);
    Node node3(0);
    Node node4(4);
    Node node5(7);
    Node node6(9);
    Node node7(3);
    Node node8(5);
    node0.pLeft = &node1;
    node0.pRight = &node2;
    node1.pLeft = &node3;
    node1.pRight = &node4;
    node2.pLeft = &node5;
    node2.pRight = &node6;
    node4.pLeft = &node7;
    node4.pRight = &node8;

    Node* pParent = LowestCommonParent(&node0, &node0, &node8);
    cout <data << endl;
    return 0;
    }

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  8. Only comparing data is not enough. I think you miss a case where two nodes has data that’re equal.

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  9. public Node lcaBST(Node current, int n1, int n2) {

    Node lca;
    if (current == null)
    return null;
    if ((n1 current.key)
    || (n1 == current.key && n2 > current.key)
    || (n2 == current.key && n1 n1 && current.key > n2)
    lca = lcaBST(current.leftChild, n1, n2);
    else if (current.key < n1&& current.key < n2)
    lca = lcaBST(current.rightChild, n1, n2);
    else
    lca = current;

    return lca;
    }</code

    VA:F [1.9.22_1171]
    0
    • When i posted this code it looks like not the whole code got posted. How do i delete my comments

      public Node lcaBST(Node current, int n1, int n2) {

      Node lca;
      if (current == null)
      return null;
      if ((n1 current.key)
      || (n1 == current.key && n2 > current.key)
      || (n2 == current.key && n1 n1 && current.key > n2)
      lca = lcaBST(current.leftChild, n1, n2);
      else if (current.key < n1&& current.key < n2)
      lca = lcaBST(current.rightChild, n1, n2);
      else
      lca = current;

      return lca;
      }

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  10. VA:F [1.9.22_1171]
    0
  11. Iterative approach:

    VA:F [1.9.22_1171]
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  12. Please note BST definition allows left <= root. So it does not work on this case:

    1. All the nodes have value 2
    2. look for the LCA 2(2) and 2(3) while 2(1) is expected but 2(0) is retruned
    2(0)
    |
    2(1)
    | |
    2(2) 2(3)

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  13. I was talking this case:

    2(0)
    |left
    2(1)
    |left
    2(2)

    looking for LCA 2(1) and 2(2). 2(1) is expected while 2(0) is returned.

    VA:F [1.9.22_1171]
    0
    • Here is my solution on this case

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  14. Hi,

    I don’t understand why the big O if this has been stated as O(n). As far as I can see, it is O(log n) as it pretty much resembles a BST search operation.

    Could someone please clarify?

    Thanks,
    Abhi.

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    0
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  16. If the binary tree can contain duplicates, then we can build the tree with the following predicate:
    n.left contains all elements n

    In this case, we must handle possibility that on of the nodes we are searching can be duplicated in the tree.

    consider the following integer valued BST:
    2
    / \
    2* 4
    /
    2
    /
    1*

    Nodes to find lca for are marked with *, i.e. 1* and 2*. Original alg will return (uppermost) 2, not 2*.

    Here is the corrected code:

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  17. If the binary tree can contain duplicates, then we can build the tree with the following predicate:
    n.left contains all elements

    n
    n.right contains all elements

    n

    In this case, we must handle possibility that on of the nodes we are searching can be duplicated in the tree.

    consider the following integer valued BST:

    Nodes to find lca for are marked with *, i.e. 1* and 2*. Original alg will return (uppermost) 2, not 2*.

    Here is the corrected code:

    VA:F [1.9.22_1171]
    0
  18. If the binary tree can contain duplicates, then we can build the tree with the following predicate:
    n.left contains all elements

    n.right contains all elements

    In this case, we must handle possibility that on of the nodes we are searching can be duplicated in the tree.

    consider the following integer valued BST:

    Nodes to find lca for are marked with *, i.e. 1* and 2*. Original alg will return (uppermost) 2, not 2*.

    Here is the corrected code:

    VA:F [1.9.22_1171]
    0
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