Question

For a given binary tree, find the largest subtree which is also binary search tree?

Example:

Input:

                   10
               /         \
             50           150
            /  \         /   \
          25    75     200    20
         / \   / \    /  \    / \
        15 35 65 30  120 135 155 250 

Output:

                   50
                  /   \
                 25   75
                / \   /
               15 35  65

Answer 1

This answer previously contained an O(n log n) algorithm based on link/cut trees. Here is a simpler O(n) solution.

The core is a procedure that accepts a node, the unique maximum BSST rooted at its left child, the unique maximum BSST rooted at its right child, and pointers to the left-most and right-most elements of these BSSTs. It destroys its inputs (avoidable with persistent data structures) and constructs the unique maximum BSST rooted at the given node, together with its minimum and maximum elements. All BSST nodes are annotated with the number of descendants. As before, this procedure is called repeatedly from a post-order traversal. To recover the sub-tree, remember the root of the largest BSST; reconstructing it requires only a simple traversal.

I’ll treat the left BSST only; the right is symmetric. If the root of the left BSST is greater than the new root, then the entire sub-tree is removed, and the new root is now left-most. Otherwise, the old left-most node is still left-most. Starting from the right-most node of the left BSST and moving upward, find the first node that is less than or equal to the root. Its right child must be removed; note now that due to the BST property, no other nodes need to go! Proceed to the root of the left BSST, updating the counts to reflect the deletion.

The reason this is O(n) is that in spite of the loop, each edge in the original tree is in essence traversed only once.

---

EDIT: collectively, the paths traversed are the maximal straight-line paths in a BST, except for the left spine and the right spine. For example, on the input

              H
             / \
            /   \
           /     \
          /       \
         /         \
        /           \
       /             \
      D               L
     / \             / \
    /   \           /   \
   /     \         /     \
  B       F       J       N
 / \     / \     / \     / \
A   C   E   G   I   K   M   O

here are the recursive calls on which each edge is traversed:

              H
             / \
            /   \
           /     \
          /       \
         /         \
        /           \
       /             \
      D               L
     / h             h \
    /   h           h   \
   /     h         h     \
  B       F       J       N
 / d     d h     h l     l \
A   C   E   G   I   K   M   O