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 traversals. 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 the 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 subarray 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 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[]
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 subarray 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;

    // construct 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 construct 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 print inorder
# traversal of a Binary Tree
def printInorder(node):
    if (node == None) :
        return
    printInorder(node.left)
    print(node.data, end = " ")
    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[]
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 subarray 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)

    Print Inorder(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)

C#

// C# program to construct full binary tree
// using its preorder traversal and preorder
// traversal of its mirror tree
using System;

class GFG
{

// A Binary Tree Node
public class Node
{
    public int data;
    public Node left, right;
};
public class INT
{
    public int a;
    public 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);
    Console.Write("{0} ", 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[]
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 subarray 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;

    // construct 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 construct 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 PrinciRaj1992 */

Javascript

<script>

// JavaScript program to construct full binary tree
// using its preorder traversal and preorder
// traversal of its mirror tree

// A Binary Tree Node
class Node
{
    constructor()
    {
        this.data = 0;
        this.left = null;
        this.right = null;
    }
};

class INT
{
    constructor(a)
    {
        this.a = a;
    }
}

// Utility function to create a new tree node
function newNode(data)
{
    var temp = new Node();
    temp.data = data;
    temp.left = temp.right = null;
    return temp;
}

// A utility function to print inorder traversal
// of a Binary Tree
function printInorder(node)
{
    if (node == null)
        return;

    printInorder(node.left);
    document.write(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[]
function conBinaryTreeUtil(pre, preM, preIndex, l, h, 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
    var root = newNode(pre[preIndex.a]);
        ++(preIndex.a);

    // If the current subarray has only one element,
    // no need to recur
    if (l == h)
        return root;

    // Search the next element of pre[] in preM[]
    var i;
    for (i = l; i <= h; ++i)
        if (pre[preIndex.a] == preM[i])
            break;

    // construct 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 construct full binary tree
// using its preorder traversal and preorder
// traversal of its mirror tree
function conBinaryTree(root,pre, preMirror, size)
{
    var preIndex = new INT(0);
    var preMIndex = 0;

    root = conBinaryTreeUtil(pre,preMirror,
                            preIndex, 0, size - 1, size);

    printInorder(root);
}

// Driver code
var preOrder = [1,2,4,5,3,6,7];
var preOrderMirror = [1,3,7,6,2,5,4];
var size = preOrder.length;
var root = new Node();
conBinaryTree(root,preOrder,preOrderMirror,size);


</script>

Output

4 2 5 1 6 3 7 

Time Complexity: O(n^2)
Auxiliary Space: O(n), The extra space is used due to the recursion call stack

• Method 2: If we observe carefully, then the reverse of the Preorder traversal of the mirror tree will be the Postorder traversal of the 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 a Full Binary Tree from given preorder and postorder traversals.