Construct a Perfect Binary Tree from Preorder Traversal

Given an array pre[], representing the Preorder traversal of a Perfect Binary Tree consisting of N nodes, the task is to construct a Perfect Binary Tree from the given Preorder Traversal and return the root of the tree.

Examples:

Input: pre[] = {1, 2, 4, 5, 3, 6, 7}
Output:
                            1
                         /    \
                      /        \
                     2          3
                  /   \       /   \ 
                /      \    /      \
              4       5   6       7

Input: pre[] = {1, 2, 3}
Output:
                            1
                         /    \
                      /        \
                    2          3

Generally to construct a binary tree, we can not do it by only using the preorder traversal, but here an extra condition is given that the binary tree is Perfect binary tree. We can use that extra condition.

For Perfect binary tree every node has either 2 or 0 children , and all the leaf nodes are present at same level. And the preorder traversal of a binary tree contains the root first, then the preorder traversal of the left subtree, then the preorder traversal of the right subtree . So for Perfect binary tree root should have same numbers of children in both subtrees , so the number ( say, n) of elements after the root in the preorder traversal should be even (2 * number of nodes in one subtree , as it is Perfect binary tree) . And since each subtrees have equal number of nodes for Perfect  binary tree, we can find the preorder traversal of the left subtree (which is half of the array after the root in preorder traversal of the whole tree), and we know the preorder traversal of the right subtree must be after the preorder traversal of the left subtree, so rest half is the preorder traversal of the right subtree.

So the first element in the preorder traversal is the root, we will build a node as the root with this element , then we can easily find the preorder traversals of the left and right subtrees of the root, and we will recursively build the left and right subtrees of the root.

Approach: The given problem can be solved using recursion. Follow the steps below to solve the problem:

• Create a function, say BuildPerfectBT_helper with parameters as preStart, preEnd, pre[] where preStart represent starting index of the array pre[] and preEnd represents the ending index of the array pre[] and perform the following steps:
  • If the value of preStart is greater than preEnd, then return NULL.
  • Initialize root as pre[preStart].
  • If the value of preStart is the same as the preEnd, then return root.
  • Initialize 4 variable, say leftPreStart as preStart + 1, rightPreStart as leftPreStart + (preEnd – leftPreStart+1)/2, leftPreEnd as rightPreStart – 1 and rightPreEnd as preEnd.
  • Modify the value of root->left by recursively calling the function buildPerfectBT_helper() with the parameters leftPreStart, leftPreEnd and pre[].
  • Modify the value of root->right by recursively calling the function buildPerfectBT_helper() with the parameters rightPreStart, rightPreEnd and pre[].
  • After performing the above steps, return root.
• After creating the Perfect Binary Tree, print the Inorder traversal of the tree.

Below is the illustration of the above steps discussed:

Step 1: build([1, 2, 4, 5, 3, 6, 7])

Step 2:
                     1
                /        \
             /            \
build([2, 4, 5])      build([3, 6, 7])

Now first element (1 here) is root, then the subarray after the first element  ( which is [2,4,5,3,6,7] here ) contains the preorder traversals of the left and right subtrees. And we know left subtree’s preorder traversal is first half , i.e, [2,4,5]  , and the right subtree’s preorder traversal is the second half , i.e, [3,6,7]. Now recursively build the left and right subtrees. 

Step 3:
                                                    1

                                   ___________|_________________

                              /                                               \

                         /                                                      \

                      2                                                          3

           ______/___________                                   _______\____________

        /                            \                               /                               \

    /                                 \                         /                                     \

build([4])                   build([5])            build([6])                          build([7]) 

Step 4: 
                   1
                /   \
             /       \
          2         3
        /   \     /   \ 
      /     \   /     \
    4       5  6     7

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Structure of the tree
struct Node {
    int data;
    Node *left, *right;

    Node(int val)
    {
        data = val;
        left = right = NULL;
    }
};

// Function to create a new node with
// the value val
Node* getNewNode(int val)
{
    Node* newNode = new Node(val);
    newNode->data = val;
    newNode->left = newNode->right = NULL;

    // Return the newly created node
    return newNode;
}

// Function to create the Perfect
// Binary Tree
Node* buildPerfectBT_helper(int preStart,
                            int preEnd,
                            int pre[])
{
    // If preStart > preEnd return NULL
    if (preStart > preEnd)
        return NULL;

    // Initialize root as pre[preStart]
    Node* root = getNewNode(pre[preStart]);
    ;

    // If the only node is left,
    // then return node
    if (preStart == preEnd)
        return root;

    // Parameters for further recursion
    int leftPreStart = preStart + 1;
    int rightPreStart = leftPreStart
                        + (preEnd - leftPreStart + 1) / 2;
    int leftPreEnd = rightPreStart - 1;
    int rightPreEnd = preEnd;

    // Recursive Call to build the
    // subtree of root node
    root->left = buildPerfectBT_helper(
        leftPreStart, leftPreEnd, pre);

    root->right = buildPerfectBT_helper(
        rightPreStart, rightPreEnd, pre);

    // Return the created root
    return root;
}

// Function to build Perfect Binary Tree
Node* buildPerfectBT(int pre[], int size)
{
    return buildPerfectBT_helper(0, size - 1, pre);
}

// Function to print the Inorder of
// the given Tree
void printInorder(Node* root)
{
    // Base Case
    if (!root)
        return;

    // Left Recursive Call
    printInorder(root->left);

    // Print the data
    cout << root->data << " ";

    // Right Recursive Call
    printInorder(root->right);
}

// Driver Code
int main()
{
    int pre[] = { 1, 2, 4, 5, 3, 6, 7 };
    int N = sizeof(pre) / sizeof(pre[0]);

    // Function Call
    Node* root = buildPerfectBT(pre, N);

    // Print Inorder Traversal
    cout << "\nInorder traversal of the tree: ";
    printInorder(root);

    return 0;
}

Java

// Java program for the above approach
public class Main
{
    // Structure of the tree
    static class Node {

        public int data;
        public Node left, right;

        public Node(int val)
        {
            data = val;
            left = right = null;
        }
    }

    // Function to create a new node with
    // the value val
    static Node getNewNode(int val)
    {
        Node newNode = new Node(val);

        // Return the newly created node
        return newNode;
    }

    // Function to create the Perfect
    // Binary Tree
    static Node buildPerfectBT_helper(int preStart, int preEnd, int[] pre)
    {
        // If preStart > preEnd return NULL
        if (preStart > preEnd)
            return null;

        // Initialize root as pre[preStart]
        Node root = getNewNode(pre[preStart]);

        // If the only node is left,
        // then return node
        if (preStart == preEnd)
            return root;

        // Parameters for further recursion
        int leftPreStart = preStart + 1;
        int rightPreStart = leftPreStart + (preEnd - leftPreStart + 1) / 2;
        int leftPreEnd = rightPreStart - 1;
        int rightPreEnd = preEnd;

        // Recursive Call to build the
        // subtree of root node
        root.left = buildPerfectBT_helper(
            leftPreStart, leftPreEnd, pre);

        root.right = buildPerfectBT_helper(
            rightPreStart, rightPreEnd, pre);

        // Return the created root
        return root;
    }

    // Function to build Perfect Binary Tree
    static Node buildPerfectBT(int[] pre, int size)
    {
        return buildPerfectBT_helper(0, size - 1, pre);
    }

    // Function to print the Inorder of
    // the given Tree
    static void printInorder(Node root)
    {
        // Base Case
        if (root == null)
            return;

        // Left Recursive Call
        printInorder(root.left);

        // Print the data
        System.out.print(root.data + " ");

        // Right Recursive Call
        printInorder(root.right);
    }

    public static void main(String[] args) {
        int[] pre = { 1, 2, 4, 5, 3, 6, 7 };
        int N = pre.length;

        // Function Call
        Node root = buildPerfectBT(pre, N);

        // Print Inorder Traversal
        System.out.print("Inorder traversal of the tree: ");
        printInorder(root);
    }
}

// This code is contributed by suresh07.

Python3

# Python3 program for the above approach

# Structure of the tree
class Node:
    def __init__(self, val):
        self.data = val
        self.left = None
        self.right = None

# Function to create a new node with
# the value val
def getNewNode(val):
    newNode = Node(val)

    # Return the newly created node
    return newNode

# Function to create the Perfect
# Binary Tree
def buildPerfectBT_helper(preStart, preEnd, pre):

    # If preStart > preEnd return NULL
    if (preStart > preEnd):
        return None

    # Initialize root as pre[preStart]
    root = getNewNode(pre[preStart])

    # If the only node is left,
    # then return node
    if (preStart == preEnd):
        return root

    # Parameters for further recursion
    leftPreStart = preStart + 1
    rightPreStart = leftPreStart + int((preEnd - leftPreStart + 1) / 2)
    leftPreEnd = rightPreStart - 1
    rightPreEnd = preEnd

    # Recursive Call to build the
    # subtree of root node
    root.left = buildPerfectBT_helper(leftPreStart, leftPreEnd, pre)

    root.right = buildPerfectBT_helper(rightPreStart, rightPreEnd, pre)

    # Return the created root
    return root

# Function to build Perfect Binary Tree
def buildPerfectBT(pre, size):
    return buildPerfectBT_helper(0, size - 1, pre)

# Function to print the Inorder of
# the given Tree
def printInorder(root):

    # Base Case
    if (root == None):
        return

    # Left Recursive Call
    printInorder(root.left)

    # Print the data
    print(root.data, "", end = "")

    # Right Recursive Call
    printInorder(root.right)

pre = [ 1, 2, 4, 5, 3, 6, 7 ]
N = len(pre)

# Function Call
root = buildPerfectBT(pre, N)

# Print Inorder Traversal
print("Inorder traversal of the tree: ", end = "")
printInorder(root)

# This code is contributed by decode2207.

C#

// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG {

    // Structure of the tree
    class Node {

        public int data;
        public Node left, right;

        public Node(int val)
        {
            data = val;
            left = right = null;
        }
    }

    // Function to create a new node with
    // the value val
    static Node getNewNode(int val)
    {
        Node newNode = new Node(val);
        // Return the newly created node
        return newNode;
    }

    // Function to create the Perfect
    // Binary Tree
    static Node buildPerfectBT_helper(int preStart, int preEnd, int[] pre)
    {
        // If preStart > preEnd return NULL
        if (preStart > preEnd)
            return null;

        // Initialize root as pre[preStart]
        Node root = getNewNode(pre[preStart]);

        // If the only node is left,
        // then return node
        if (preStart == preEnd)
            return root;

        // Parameters for further recursion
        int leftPreStart = preStart + 1;
        int rightPreStart = leftPreStart + (preEnd - leftPreStart + 1) / 2;
        int leftPreEnd = rightPreStart - 1;
        int rightPreEnd = preEnd;

        // Recursive Call to build the
        // subtree of root node
        root.left = buildPerfectBT_helper(
            leftPreStart, leftPreEnd, pre);

        root.right = buildPerfectBT_helper(
            rightPreStart, rightPreEnd, pre);

        // Return the created root
        return root;
    }

    // Function to build Perfect Binary Tree
    static Node buildPerfectBT(int[] pre, int size)
    {
        return buildPerfectBT_helper(0, size - 1, pre);
    }

    // Function to print the Inorder of
    // the given Tree
    static void printInorder(Node root)
    {
        // Base Case
        if (root == null)
            return;

        // Left Recursive Call
        printInorder(root.left);

        // Print the data
        Console.Write(root.data + " ");

        // Right Recursive Call
        printInorder(root.right);
    }

  static void Main() {
    int[] pre = { 1, 2, 4, 5, 3, 6, 7 };
    int N = pre.Length;

    // Function Call
    Node root = buildPerfectBT(pre, N);

    // Print Inorder Traversal
    Console.Write("Inorder traversal of the tree: ");
    printInorder(root);
  }
}

// This code is contributed by mukesh07.

Javascript

<script>
    // Javascript program for the above approach

    // Structure of the tree
    class Node
    {
        constructor(val) {
           this.left = null;
           this.right = null;
           this.data = val;
        }
    }

    // Function to create a new node with
    // the value val
    function getNewNode(val)
    {
        let newNode = new Node(val);
        // Return the newly created node
        return newNode;
    }

    // Function to create the Perfect
    // Binary Tree
    function buildPerfectBT_helper(preStart, preEnd, pre)
    {
        // If preStart > preEnd return NULL
        if (preStart > preEnd)
            return null;

        // Initialize root as pre[preStart]
        let root = getNewNode(pre[preStart]);

        // If the only node is left,
        // then return node
        if (preStart == preEnd)
            return root;

        // Parameters for further recursion
        let leftPreStart = preStart + 1;
        let rightPreStart = leftPreStart
                            + parseInt((preEnd - leftPreStart + 1) / 2);
        let leftPreEnd = rightPreStart - 1;
        let rightPreEnd = preEnd;

        // Recursive Call to build the
        // subtree of root node
        root.left = buildPerfectBT_helper(
            leftPreStart, leftPreEnd, pre);

        root.right = buildPerfectBT_helper(
            rightPreStart, rightPreEnd, pre);

        // Return the created root
        return root;
    }

    // Function to build Perfect Binary Tree
    function buildPerfectBT(pre, size)
    {
        return buildPerfectBT_helper(0, size - 1, pre);
    }

    // Function to print the Inorder of
    // the given Tree
    function printInorder(root)
    {
        // Base Case
        if (root == null)
            return;

        // Left Recursive Call
        printInorder(root.left);

        // Print the data
        document.write(root.data + " ");

        // Right Recursive Call
        printInorder(root.right);
    }

    let pre = [ 1, 2, 4, 5, 3, 6, 7 ];
    let N = pre.length;

    // Function Call
    let root = buildPerfectBT(pre, N);

    // Print Inorder Traversal
    document.write("Inorder traversal of the tree: ");
    printInorder(root);

// This code is contributed by divyesh072019.
</script>

Output:

Inorder traversal of the tree: 4 2 5 1 6 3 7

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