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# Post Order Traversal of Binary Tree in O(N) using O(1) space

Prerequisites:- Morris Inorder Traversal, Tree Traversals (Inorder, Preorder and Postorder)
Given a Binary Tree, the task is to print the elements in post order using O(N) time complexity and constant space.

Input:   1
/   \
2       3
/ \     / \
4   5   6   7
/ \
8   9
Output: 8 9 4 5 2 6 7 3 1

Input:   5
/   \
7       3
/ \     / \
4   11  13  9
/ \
8   4
Output: 8 4 4 11 7 13 9 3 5

Method 1: Using Morris Inorder Traversal

1. Create a dummy node and make the root as it’s left child.
2. Initialize current with dummy node.
3. While current is not NULL
• If the current does not have a left child traverse the right child, current = current->right
• Otherwise,
1. Find the rightmost child in the left subtree.
2. If rightmost child’s right child is NULL
• Make current as the right child of the rightmost node.
• Traverse the left child, current = current->left
3. Otherwise,
• Set the rightmost child’s right pointer to NULL.
• From current’s left child, traverse along with the right children until the rightmost child and reverse the pointers.
• Traverse back from rightmost child to current’s left child node by reversing the pointers and printing the elements.
• Traverse the right child, current = current->right

Below is the diagram showing the rightmost child in the left subtree, pointing to its inorder successor.

Below is the diagram which highlights the path 1->2->5->9 and the way the nodes are processed and printed as per the above algorithm.

Below is the implementation of the above approach:

## C++

 // C++ program to implement// Post Order traversal// of Binary Tree in O(N)// time and O(1) space#include using namespace std; class node{    public:    int data;    node *left, *right;}; // Helper function that allocates a// new node with the given data and// NULL left and right pointers.node* newNode(int data){    node* temp = new node();    temp->data = data;    temp->left = temp->right = NULL;    return temp;} // Postorder traversal without recursion// and without stackvoid postOrderConstSpace(node* root){    if (root == NULL)            return;         node* current = newNode(-1);        node* pre = NULL;        node* prev = NULL;        node* succ = NULL;        node* temp = NULL;                 current->left = root;             while (current)    {                 // If left child is null.        // Move to right child.        if (current->left == NULL)        {            current = current->right;        }        else        {            pre = current->left;                         // Inorder predecessor            while (pre->right &&                pre->right != current)                pre = pre->right;                         // The connection between current and            // predecessor is made            if (pre->right == NULL)            {                                 // Make current as the right                // child of the right most node                pre->right = current;                                 // Traverse the left child                current = current->left;            }            else            {                pre->right = NULL;                succ = current;                current = current->left;                prev = NULL;                                 // Traverse along the right                // subtree to the                // right-most child                while (current != NULL)                {                    temp = current->right;                    current->right = prev;                    prev = current;                    current = temp;                }                                 // Traverse back                // to current's left child                // node                while (prev != NULL)                {                    cout << prev->data << " ";                    temp = prev->right;                    prev->right = current;                    current = prev;                    prev = temp;                }                 current = succ;                current = current->right;            }        }    }} // Driver codeint main(){   /* Constructed tree is as follows:-                      1                   /     \                  2       3                 / \     / \                4   5   6   7                   / \                  8   9      */    node* root = NULL;     root = newNode(1);    root->left = newNode(2);    root->right = newNode(3);     root->left->left = newNode(4);    root->left->right = newNode(5);     root->right->left = newNode(6);    root->right->right = newNode(7);     root->left->right->left = newNode(8);    root->left->right->right = newNode(9);         postOrderConstSpace(root);    return 0;} // This code is contributed by Saurav Chaudhary

## Java

 // Java program to implement// Post Order traversal// of Binary Tree in O(N)// time and O(1) space // Definition of the// binary treeclass TreeNode {    public int data;    public TreeNode left;    public TreeNode right;    public TreeNode(int data)    {        this.data = data;    }     public String toString()    {        return data + " ";    }} public class PostOrder {     TreeNode root;     // Function to find Post Order    // Traversal Using Constant space    void postOrderConstantspace(TreeNode                                    root)    {        if (root == null)            return;         TreeNode current            = new TreeNode(-1),            pre = null;        TreeNode prev = null,                succ = null,                temp = null;        current.left = root;         while (current != null) {             // Go to the right child            // if current does not            // have a left child             if (current.left == null) {                current = current.right;            }             else {                 // Traverse left child                pre = current.left;                 // Find the right most child                // in the left subtree                while (pre.right != null                    && pre.right != current)                    pre = pre.right;                 if (pre.right == null) {                     // Make current as the right                    // child of the right most node                    pre.right = current;                     // Traverse the left child                    current = current.left;                }                 else {                    pre.right = null;                    succ = current;                    current = current.left;                    prev = null;                     // Traverse along the right                    // subtree to the                    // right-most child                     while (current != null) {                        temp = current.right;                        current.right = prev;                        prev = current;                        current = temp;                    }                     // Traverse back from                    // right most child to                    // current's left child node                     while (prev != null) {                         System.out.print(prev);                        temp = prev.right;                        prev.right = current;                        current = prev;                        prev = temp;                    }                     current = succ;                    current = current.right;                }            }        }    }     // Driver Code    public static void main(String[] args)    {        /* Constructed tree is as follows:-                      1                    /   \                   2     3                  / \    / \                 4   5   6  7                    / \                    8  9        */        PostOrder tree = new PostOrder();        tree.root = new TreeNode(1);        tree.root.left = new TreeNode(2);        tree.root.right = new TreeNode(3);        tree.root.left.left = new TreeNode(4);        tree.root.left.right            = new TreeNode(5);        tree.root.right.left            = new TreeNode(6);        tree.root.right.right            = new TreeNode(7);        tree.root.left.right.left            = new TreeNode(8);        tree.root.left.right.right            = new TreeNode(9);         tree.postOrderConstantspace(            tree.root);    }}

## Python3

 # Python3 program to implement# Post Order traversal# of Binary Tree in O(N)# time and O(1) spaceclass node:         def __init__(self, data):                 self.data = data        self.left = None        self.right = None # Helper function that allocates a# new node with the given data and# None left and right pointers.def newNode(data):     temp = node(data)    return temp # Postorder traversal without recursion# and without stackdef postOrderConstSpace(root):     if (root == None):        return     current = newNode(-1)    pre = None    prev = None    succ = None    temp = None         current.left = root             while (current):                 # If left child is None.        # Move to right child.        if (current.left == None):            current = current.right        else:            pre = current.left                         # Inorder predecessor            while (pre.right and                   pre.right != current):                pre = pre.right                         # The connection between current            # and predecessor is made            if (pre.right == None):                                 # Make current as the right                # child of the right most node                pre.right = current                                 # Traverse the left child                current = current.left                         else:                             pre.right = None                succ = current                current = current.left                prev = None                                 # Traverse along the right                # subtree to the                # right-most child                while (current != None):                    temp = current.right                    current.right = prev                    prev = current                    current = temp                             # Traverse back                # to current's left child                # node                while (prev != None):                    print(prev.data, end = ' ')                    temp = prev.right                    prev.right = current                    current = prev                    prev = temp                 current = succ                current = current.right # Driver codeif __name__=='__main__':         ''' Constructed tree is as follows:-                       1                    /     \                   2      3                  / \     / \                 4   5  6   7                    / \                   8   9        '''    root = None     root = newNode(1)    root.left = newNode(2)    root.right = newNode(3)     root.left.left = newNode(4)    root.left.right = newNode(5)     root.right.left = newNode(6)    root.right.right = newNode(7)     root.left.right.left = newNode(8)    root.left.right.right = newNode(9)         postOrderConstSpace(root) # This code is contributed by pratham76

## C#

 // C# program to implement// Post Order traversal// of Binary Tree in O(N)// time and O(1) spaceusing System; // Definition of the// binary treepublic class  TreeNode{    public int data;    public TreeNode left, right;       public TreeNode(int item)    {        data = item;        left = right = null;    }} class PostOrder{     public TreeNode root; // Function to find Post Order// Traversal Using Constant spacevoid postOrderConstantspace(TreeNode root){    if (root == null)        return;       TreeNode current = new TreeNode(-1), pre = null;    TreeNode prev = null,             succ = null,             temp = null;                  current.left = root;     while (current != null)    {                 // Go to the right child        // if current does not        // have a left child        if (current.left == null)        {            current = current.right;        }         else        {                         // Traverse left child            pre = current.left;             // Find the right most child            // in the left subtree            while (pre.right != null &&                   pre.right != current)                pre = pre.right;             if (pre.right == null)            {                                 // Make current as the right                // child of the right most node                pre.right = current;                 // Traverse the left child                current = current.left;            }            else            {                pre.right = null;                succ = current;                current = current.left;                prev = null;                 // Traverse along the right                // subtree to the                // right-most child                while (current != null)                {                    temp = current.right;                    current.right = prev;                    prev = current;                    current = temp;                }                 // Traverse back from                // right most child to                // current's left child node                while (prev != null)                {                   Console.Write(prev.data + " ");                    temp = prev.right;                    prev.right = current;                    current = prev;                    prev = temp;                }                current = succ;                current = current.right;            }        }    }} // Driver codestatic public void Main (){         /* Constructed tree is as follows:-                      1                   /     \                  2       3                 / \     / \                4   5   6   7                   / \                  8   9      */    PostOrder tree = new PostOrder();    tree.root = new TreeNode(1);    tree.root.left = new TreeNode(2);    tree.root.right = new TreeNode(3);    tree.root.left.left = new TreeNode(4);    tree.root.left.right = new TreeNode(5);    tree.root.right.left = new TreeNode(6);    tree.root.right.right = new TreeNode(7);    tree.root.left.right.left = new TreeNode(8);    tree.root.left.right.right = new TreeNode(9);     tree.postOrderConstantspace(tree.root);}} // This code is contributed by offbeat

## Javascript



Output

4 8 9 5 2 6 7 3 1

Time Complexity: O(N)
Auxiliary Space: O(1)

Method 2: In method 1, we traverse a path, reverse references, print nodes as we restore the references by reversing them again. In method 2, instead of reversing paths and restoring the structure, we traverse to the parent node from the current node using the current node’s left subtree. This could be faster depending on the tree structure, for example in a right-skewed tree.

The following algorithm and diagrams provide the details of the approach.

Below is the conceptual diagram showing how the left and right child references are used to traverse back and forth.

Below is the diagram which highlights the path 1->2->5->9 and the way the nodes are processed and printed as per the above algorithm.

Below is the implementation of the above approach:

## C++

 // C++ Program to implement the above approach#include using namespace std; struct TreeNode {    TreeNode* left;    TreeNode* right;    int data;      TreeNode(int data)    {        this->data = data;        this->left = nullptr;        this->right = nullptr;    }}; TreeNode* root;    // Function to Calculate Post// Order Traversal Using// Constant Spacestatic void postOrderConstantspace(TreeNode* root){  if (root == nullptr)    return;   TreeNode* current = nullptr;  TreeNode* prevNode = nullptr;  TreeNode* pre = nullptr;  TreeNode* ptr = nullptr;  TreeNode* netChild = nullptr;  TreeNode* prevPtr = nullptr;   current = root;   while (current != nullptr)  {    if (current->left == nullptr)    {      current->left = prevNode;       // Set prevNode to current      prevNode = current;      current = current->right;    }    else    {      pre = current->left;       // Find the right most child      // in the left subtree      while (pre->right != nullptr &&             pre->right != current)        pre = pre->right;       if (pre->right == nullptr)      {        pre->right = current;        current = current->left;      }      else      {        // Set the right most        // child's right pointer        // to NULL        pre->right = nullptr;        cout << pre->data << " ";        ptr = pre;        netChild = pre;        prevPtr = pre;         while (ptr != nullptr)        {          if (ptr->right == netChild)          {            cout << ptr->data << " ";            netChild = ptr;            prevPtr->left = nullptr;          }           if (ptr == current->left)            break;           // Break the loop          // all the left subtree          // nodes of current          // processed          prevPtr = ptr;          ptr = ptr->left;        }         prevNode = current;        current = current->right;      }    }  }   cout << prevNode->data << " ";   // Last path traversal  // that includes the root.  ptr = prevNode;  netChild = prevNode;  prevPtr = prevNode;   while (ptr != nullptr)  {    if (ptr->right == netChild)    {      cout << ptr->data << " ";      netChild = ptr;      prevPtr->left = nullptr;    }    if (ptr == root)      break;     prevPtr = ptr;    ptr = ptr->left;  }} int main(){    /* Constructed tree is as follows:-                      1                   /     \                  2       3                 / \     / \                4   5   6   7                   / \                  8   9      */  root = new TreeNode(1);  root->left = new TreeNode(2);  root->right = new TreeNode(3);  root->left->left = new TreeNode(4);  root->left->right = new TreeNode(5);  root->right->left = new TreeNode(6);  root->right->right = new TreeNode(7);  root->left->right->left = new TreeNode(8);  root->left->right->right = new TreeNode(9);  postOrderConstantspace(root);     return 0;} // This code is contributed by mukesh07.

## Java

 // Java Program to implement// the above approachclass TreeNode {    public int data;    public TreeNode left;    public TreeNode right;     public TreeNode(int data)    {        this.data = data;    }     public String toString()    {        return data + " ";    }} public class PostOrder {    TreeNode root;     // Function to Calculate Post    // Order Traversal    // Using Constant Space    void postOrderConstantspace(TreeNode root)    {        if (root == null)            return;         TreeNode current = null;        TreeNode prevNode = null;        TreeNode pre = null;        TreeNode ptr = null;        TreeNode netChild = null;        TreeNode prevPtr = null;         current = root;        while (current != null) {            if (current.left == null) {                current.left = prevNode;                // Set prevNode to current                prevNode = current;                current = current.right;            }            else {                pre = current.left;                // Find the right most child                // in the left subtree                while (pre.right != null                    && pre.right != current)                    pre = pre.right;                 if (pre.right == null) {                    pre.right = current;                    current = current.left;                }                else {                    // Set the right most                    // child's right pointer                    // to NULL                    pre.right = null;                    System.out.print(pre);                     ptr = pre;                    netChild = pre;                    prevPtr = pre;                    while (ptr != null) {                        if (ptr.right == netChild) {                            System.out.print(ptr);                            netChild = ptr;                            prevPtr.left = null;                        }                         if (ptr == current.left)                            break;                        // Break the loop                        // all the left subtree                        // nodes of current                        // processed                         prevPtr = ptr;                        ptr = ptr.left;                    }                     prevNode = current;                    current = current.right;                }            }        }         System.out.print(prevNode);         // Last path traversal        // that includes the root.        ptr = prevNode;        netChild = prevNode;        prevPtr = prevNode;        while (ptr != null) {            if (ptr.right == netChild) {                System.out.print(ptr);                netChild = ptr;                prevPtr.left = null;            }            if (ptr == root)                break;             prevPtr = ptr;            ptr = ptr.left;        }    }     // Main Function    public static void main(String[] args)    {        /* Constructed tree is as follows:-                      1                   /     \                  2       3                 / \     / \                4   5   6   7                   / \                  8   9      */        PostOrder tree = new PostOrder();        tree.root = new TreeNode(1);        tree.root.left = new TreeNode(2);        tree.root.right = new TreeNode(3);        tree.root.left.left            = new TreeNode(4);        tree.root.left.right            = new TreeNode(5);        tree.root.right.left            = new TreeNode(6);        tree.root.right.right            = new TreeNode(7);        tree.root.left.right.left            = new TreeNode(8);        tree.root.left.right.right            = new TreeNode(9);         tree.postOrderConstantspace(            tree.root);    }}

## Python3

 # Python3 Program to implement the above approachclass TreeNode:    def __init__(self, data):        self.data = data        self.left = None        self.right = None # Function to Calculate Post# Order Traversal Using# Constant Spacedef postOrderConstantspace(root):  if root == None:    return   current = None  prevNode = None  pre = None  ptr = None  netChild = None  prevPtr = None   current = root   while current != None:    if current.left == None:      current.left = prevNode       # Set prevNode to current      prevNode = current      current = current.right    else:      pre = current.left             # Find the right most child      # in the left subtree      while pre.right != None and pre.right != current:        pre = pre.right       if pre.right == None:        pre.right = current        current = current.left      else:        # Set the right most        # child's right pointer        # to NULL        pre.right = None        print(pre.data, end = " ")        ptr = pre        netChild = pre        prevPtr = pre         while ptr != None:          if ptr.right == netChild:            print(ptr.data, end = " ")            netChild = ptr            prevPtr.left = None           if ptr == current.left:            break           # Break the loop          # all the left subtree          # nodes of current          # processed          prevPtr = ptr          ptr = ptr.left         prevNode = current        current = current.right   print(prevNode.data, end = " ")   # Last path traversal  # that includes the root.  ptr = prevNode  netChild = prevNode  prevPtr = prevNode   while ptr != None:    if ptr.right == netChild:      print(ptr.data, end = " ")      netChild = ptr      prevPtr.left = None           if (ptr == root):      break     prevPtr = ptr    ptr = ptr.left """ Constructed tree is as follows:-                  1               /     \              2       3             / \     / \            4   5   6   7               / \              8   9"""root = TreeNode(1)root.left = TreeNode(2)root.right = TreeNode(3)root.left.left = TreeNode(4)root.left.right = TreeNode(5)root.right.left = TreeNode(6)root.right.right = TreeNode(7)root.left.right.left = TreeNode(8)root.left.right.right = TreeNode(9)postOrderConstantspace(root) # This code is contributed by divyeshrabadiya07.

## C#

 // C# Program to implement// the above approachusing System;class TreeNode{     public int data;public TreeNode left;public TreeNode right;  public TreeNode(int data){  this.data = data;}  public string toString(){  return data + " ";}}  class PostOrder{     TreeNode root;  // Function to Calculate Post// Order Traversal Using// Constant Spacevoid postOrderConstantspace(TreeNode root){  if (root == null)    return;   TreeNode current = null;  TreeNode prevNode = null;  TreeNode pre = null;  TreeNode ptr = null;  TreeNode netChild = null;  TreeNode prevPtr = null;   current = root;     while (current != null)  {    if (current.left == null)    {      current.left = prevNode;             // Set prevNode to current      prevNode = current;      current = current.right;    }    else    {      pre = current.left;             // Find the right most child      // in the left subtree      while (pre.right != null &&             pre.right != current)        pre = pre.right;       if (pre.right == null)      {        pre.right = current;        current = current.left;      }      else      {        // Set the right most        // child's right pointer        // to NULL        pre.right = null;        Console.Write(pre.data + " ");        ptr = pre;        netChild = pre;        prevPtr = pre;                 while (ptr != null)        {          if (ptr.right == netChild)          {            Console.Write(ptr.data + " ");            netChild = ptr;            prevPtr.left = null;          }           if (ptr == current.left)            break;                     // Break the loop          // all the left subtree          // nodes of current          // processed          prevPtr = ptr;          ptr = ptr.left;        }         prevNode = current;        current = current.right;      }    }  }   Console.Write(prevNode.data + " ");   // Last path traversal  // that includes the root.  ptr = prevNode;  netChild = prevNode;  prevPtr = prevNode;     while (ptr != null)  {    if (ptr.right == netChild)    {      Console.Write(ptr.data + " ");      netChild = ptr;      prevPtr.left = null;    }    if (ptr == root)      break;     prevPtr = ptr;    ptr = ptr.left;  }}  // Driver codepublic static void Main(string[] args){  /* Constructed tree is as follows:-                      1                   /     \                  2       3                 / \     / \                4   5   6   7                   / \                  8   9      */  PostOrder tree = new PostOrder();  tree.root = new TreeNode(1);  tree.root.left = new TreeNode(2);  tree.root.right = new TreeNode(3);  tree.root.left.left = new TreeNode(4);  tree.root.left.right = new TreeNode(5);  tree.root.right.left = new TreeNode(6);  tree.root.right.right = new TreeNode(7);  tree.root.left.right.left = new TreeNode(8);  tree.root.left.right.right = new TreeNode(9);  tree.postOrderConstantspace(tree.root);}} // This code is contributed by Rutvik_56

## Javascript



Output

4 8 9 5 2 6 7 3 1

Time Complexity: O(N)
Auxiliary Space: O(1)