Morris traversal for Postorder

Morris Traversal is an efficient method for traversing binary trees without using any additional space for stacks or recursion. In this specific case, the postorder traversal of the binary tree is achieved using the Morris Traversal technique.

Problem Statement and Description

Given a binary tree, the goal is to traverse its nodes in a postorder sequence using the Morris Traversal technique. Postorder traversal involves visiting the left subtree, then the right subtree, and finally the root node.

Example

Consider the following binary tree:

      4
    /   \
   8     3
  / \   / \
 1   6 7   2
    /   \   \
   9     10  11

The goal is to perform postorder traversal on this tree using Morris Traversal.

Idea to Solve

The Morris Traversal algorithm utilizes threaded binary trees to achieve efficient traversal without using additional memory. In your case, for postorder traversal, the key idea is to reverse the right pointers of nodes while traversing the tree in a modified form of the pre-order traversal. This allows you to visit nodes in the required order while maintaining the tree's original structure.

Pseudocode

Here's a high-level pseudocode representation of the algorithm:

class TreeNode
    properties: data, left, right

class BinaryTree
    properties: root

function morrisPostorder
    if root is null
        return

    create a dummy node with value 0 and set its left child to root
    current = dummy_node
    while current is not null
        if current.left is null
            // No left child, move to right child
            current = current.right
        else
            // Find the rightmost node in the left subtree
            auxiliary = current.left
            while auxiliary.right is not null and auxiliary.right is not current
                auxiliary = auxiliary.right
            
            if auxiliary.right is null
                // Set right pointer to current and move to left child
                auxiliary.right = current
                current = current.left
            else
                // Reverse right pointers and traverse back from rightmost node
                print_nodes_and_reverse(current.left, auxiliary)
                auxiliary.right = null
                current = current.right
    
    print result (postorder traversal)

function print_nodes_and_reverse(from_node, to_node)
    reverse_right_pointers(from_node, to_node)
    current = to_node
    while true
        print current.data
        if current is from_node
            break
        current = current.right

function reverse_right_pointers(from_node, to_node)
    if from_node is to_node
        return
    current = from_node
    prev = null
    while current is not to_node
        next = current.right
        current.right = prev
        prev = current
        current = next

function main
    create a new BinaryTree called "tree"
    // Construct tree here

    print "Morris Postorder:"
    tree.morrisPostorder()

Algorithm Explanation

The algorithm starts by initializing the dummy_node to aid in traversal. It then iterates through the tree while reversing and restoring the right pointers of nodes as needed. When encountering a node with no left child, it moves to the right child. When encountering a node with a left child, it finds the rightmost node of the left subtree, reverses the right pointers from the left child to the rightmost node, and then traverses back, printing nodes in the postorder sequence.

Code Solution

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Time Complexity

The Morris Traversal algorithm has a time complexity of O(n), where n is the number of nodes in the binary tree. This is because each node is visited at most twice: once when traversing down the tree and once when traversing back up. The algorithm avoids the need for an explicit stack, reducing the space complexity to O(1).