Construct Full Binary Tree using its Preorder traversal and Preorder traversal of its mirror tree

Given two arrays that represent Preorder traversals of a full binary tree and its mirror tree, we need to write a program to construct the binary tree using these two Preorder traversals.

A Full Binary Tree is a binary tree where every node has either 0 or 2 children.

Note: It is not possible to construct a general binary tree using these two traversal. But we can create a full binary tree using the above traversals without any ambiguity. For more details refer to this article.

Examples:

Input :  preOrder[] = {1,2,4,5,3,6,7}
         preOrderMirror[] = {1,3,7,6,2,5,4}

Output :          1
               /    
              2      3
            /      /  
           4     5 6    7

• Method 1: Let us consider the two given arrays as preOrder[] = {1, 2, 4, 5, 3, 6, 7} and preOrderMirror[] = {1 ,3 ,7 ,6 ,2 ,5 ,4}.
  In both preOrder[] and preOrderMirror[], the leftmost element is root of tree. Since the tree is full and array size is more than 1. The value next to 1 in preOrder[], must be left child of root and value next to 1 in preOrderMirror[] must be right child of root. So we know 1 is root and 2 is left child and 3 is the right child. How to find the all nodes in left subtree? We know 2 is root of all nodes in left subtree and 3 is root of all nodes in right subtree. All nodes from and 2 in preOrderMirror[] must be in left subtree of root node 1 and all node after 3 and before 2 in preOrderMirror[] must be in right subtree of root node 1. Now we know 1 is root, elements {2, 5, 4} are in left subtree, and the elements {3, 7, 6} are in the right subtree.

             1
          /    
         /      
      {2,5,4}  {3,7,6}
  

  We will recursively follow the above approach and get the below tree:

                    1
                 /    
                2      3
              /      /  
             4     5 6    7
  

  Below is the implementation of above approach:
  

  C++

  // C++ program to construct full binary tree // using its preorder traversal and preorder // traversal of its mirror tree    #include<bits/stdc++.h> using namespace std;    // A Binary Tree Node struct Node {     int data;     struct Node *left, *right; };    // Utility function to create a new tree node Node* newNode(int data) {     Node *temp = new Node;     temp->data = data;     temp->left = temp->right = NULL;     return temp; }    // A utility function to print inorder traversal  // of a Binary Tree void printInorder(Node* node) {     if (node == NULL)         return;        printInorder(node->left);     printf("%d ", node->data);     printInorder(node->right); }    // A recursive function to construct Full binary tree //  from pre[] and preM[]. preIndex is used to keep  // track of index in pre[]. l is low index and h is high  //index for the current subarray in preM[] Node* constructBinaryTreeUtil(int pre[], int preM[],                     int &preIndex, int l,int h,int size) {         // Base case     if (preIndex >= size || l > h)         return NULL;        // The first node in preorder traversal is root.      // So take the node at preIndex from preorder and      // make it root, and increment preIndex     Node* root = newNode(pre[preIndex]);         ++(preIndex);        // If the current subarry has only one element,      // no need to recur     if (l == h)         return root;            // Search the next element of pre[] in preM[]     int i;     for (i = l; i <= h; ++i)         if (pre[preIndex] == preM[i])             break;        // construct left and right subtrees recursively         if (i <= h)     {         root->left = constructBinaryTreeUtil (pre, preM,                                      preIndex, i, h, size);         root->right = constructBinaryTreeUtil (pre, preM,                                   preIndex, l+1, i-1, size);     }          // return root     return root;     }    // function to construct full binary tree // using its preorder traversal and preorder // traversal of its mirror tree void constructBinaryTree(Node* root,int pre[],                         int preMirror[], int size) {     int preIndex = 0;     int preMIndex = 0;        root =  constructBinaryTreeUtil(pre,preMirror,                             preIndex,0,size-1,size);        printInorder(root); }    // Driver program to test above functions int main() {     int preOrder[] = {1,2,4,5,3,6,7};     int preOrderMirror[] = {1,3,7,6,2,5,4};        int size = sizeof(preOrder)/sizeof(preOrder[0]);        Node* root = new Node;         constructBinaryTree(root,preOrder,preOrderMirror,size);        return 0; }

  Java

  // Java program to construct full binary tree  // using its preorder traversal and preorder  // traversal of its mirror tree  class GFG {    // A Binary Tree Node  static class Node  {      int data;      Node left, right;  };  static class INT {     int a;     INT(int a){this.a=a;} }    // Utility function to create a new tree node  static Node newNode(int data)  {      Node temp = new Node();      temp.data = data;      temp.left = temp.right = null;      return temp;  }     // A utility function to print inorder traversal  // of a Binary Tree  static void printInorder(Node node)  {      if (node == null)          return;         printInorder(node.left);      System.out.printf("%d ", node.data);      printInorder(node.right);  }     // A recursive function to con Full binary tree  // from pre[] and preM[]. preIndex is used to keep  // track of index in pre[]. l is low index and h is high  //index for the current subarray in preM[]  static Node conBinaryTreeUtil(int pre[], int preM[],                      INT preIndex, int l, int h, int size)  {      // Base case      if (preIndex.a >= size || l > h)          return null;         // The first node in preorder traversal is root.      // So take the node at preIndex from preorder and      // make it root, and increment preIndex      Node root = newNode(pre[preIndex.a]);          ++(preIndex.a);         // If the current subarry has only one element,      // no need to recur      if (l == h)          return root;             // Search the next element of pre[] in preM[]      int i;      for (i = l; i <= h; ++i)          if (pre[preIndex.a] == preM[i])              break;         // con left and right subtrees recursively      if (i <= h)      {          root.left = conBinaryTreeUtil (pre, preM,                                      preIndex, i, h, size);          root.right = conBinaryTreeUtil (pre, preM,                                  preIndex, l + 1, i - 1, size);      }         // return root      return root;      }     // function to con full binary tree  // using its preorder traversal and preorder  // traversal of its mirror tree  static void conBinaryTree(Node root,int pre[],                          int preMirror[], int size)  {      INT preIndex = new INT(0);      int preMIndex = 0;         root = conBinaryTreeUtil(pre,preMirror,                              preIndex, 0, size - 1, size);         printInorder(root);  }     // Driver code public static void main(String args[]) {      int preOrder[] = {1,2,4,5,3,6,7};      int preOrderMirror[] = {1,3,7,6,2,5,4};         int size = preOrder.length;         Node root = new Node();         conBinaryTree(root,preOrder,preOrderMirror,size);  }  }     // This code is contributed by Arnab Kundu

  Python3

  # Python3 program to construct full binary  # tree using its preorder traversal and  # preorder traversal of its mirror tree     # Utility function to create a new tree node  class newNode:     def __init__(self,data):         self.data = data         self.left = self.right = None    # A utility function to pr inorder # traversal of a Binary Tree  def prInorder(node):      if (node == None) :         return     prInorder(node.left)      print(node.data, end = " ")      prInorder(node.right)     # A recursive function to construct Full   # binary tree from pre[] and preM[].  # preIndex is used to keep track of index  # in pre[]. l is low index and h is high  # index for the current subarray in preM[]  def constructBinaryTreeUtil(pre, preM, preIndex,                                     l, h, size):      # Base case      if (preIndex >= size or l > h) :         return None , preIndex        # The first node in preorder traversal       # is root. So take the node at preIndex      # from preorder and make it root, and      # increment preIndex      root = newNode(pre[preIndex])      preIndex += 1        # If the current subarry has only      # one element, no need to recur      if (l == h):          return root, preIndex        # Search the next element of      # pre[] in preM[]     i = 0     for i in range(l, h + 1):          if (pre[preIndex] == preM[i]):                  break        # construct left and right subtrees     # recursively      if (i <= h):             root.left, preIndex = constructBinaryTreeUtil(pre, preM, preIndex,                                                                i, h, size)          root.right, preIndex = constructBinaryTreeUtil(pre, preM, preIndex,                                                         l + 1, i - 1, size)         # return root      return root, preIndex    # function to construct full binary tree  # using its preorder traversal and preorder  # traversal of its mirror tree def constructBinaryTree(root, pre, preMirror, size):         preIndex = 0     preMIndex = 0        root, x = constructBinaryTreeUtil(pre, preMirror, preIndex,                                               0, size - 1, size)         prInorder(root)     # Driver code  if __name__ =="__main__":        preOrder = [1, 2, 4, 5, 3, 6, 7]     preOrderMirror = [1, 3, 7, 6, 2, 5, 4]        size = 7     root = newNode(0)         constructBinaryTree(root, preOrder,                          preOrderMirror, size)     # This code is contributed by # Shubham Singh(SHUBHAMSINGH10)

  Output:

  4 2 5 1 6 3 7 
  

• Method 2: If we observe carefully, then reverse of the Preorder traversal of mirror tree will be the Postorder traversal of original tree. We can construct the tree from given Preorder and Postorder traversals in a similar manner as above. You can refer to this article on how to Construct Full Binary Tree from given preorder and postorder traversals.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.