Question: I know the theritical definition for distributed computing and interproces communication.

But in real time I was not able to come to conclusion that when we go for distributed or interprocess.

Tell me some scenario where we can go for distributed computing or interprocess communication by example.

Answer: 
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

```
```
GetLargestSortedBinarySubtree(thisNode, ref OverallBestTree)
    if thisNode == null
        Return null
    LeftLargest = GetLargestSortedBinarySubtree(thisNode.LeftNode, ref OverallBestTree)
    RightLargest = GetLargestSortedBinarySubtree(thisNode.RightNode, ref OverallBestTree)
    if LeftLargest.Max < thisNode.Value & RightLargest.Min > thisNode.Value
        currentBestTree = new BinaryTree(LeftLargest, thisNode.Value, RightLargest)
    else if LeftLargest.Max < thisNode.Value
        currentBestTree = new BinaryTree(LeftLargest, thisNode.Value, null)
    else if RightLargest.Min > thisNode.Value
        currentBestTree = new BinaryTree(null, thisNode.Value, RightLargest)
    else
        currentBestTree = new BinaryTree(null, thisNode.Value, null)
    if (currentBestTree.Size > OverallBestTree.Size)
        OverallBestTree = currentBestTree
    return currentBestTree

```

As BlueRaja pointed out, this algorithm is not correct.

It should really be called `GetLargestSortedBinarySubtreeThatCanBeRecursivelyConstructedFromMaximalSortedSubtrees`.