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 7Input: 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 treestruct Node { int data; Node *left, *right; Node(int val) { data = val; left = right = NULL; }};// Function to create a new node with// the value valNode* 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 TreeNode* 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 TreeNode* buildPerfectBT(int pre[], int size){ return buildPerfectBT_helper(0, size - 1, pre);}// Function to print the Inorder of// the given Treevoid 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 Codeint 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 approachpublic 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 treeclass Node: def __init__(self, val): self.data = val self.left = None self.right = None# Function to create a new node with# the value valdef getNewNode(val): newNode = Node(val) # Return the newly created node return newNode# Function to create the Perfect# Binary Treedef 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 Treedef buildPerfectBT(pre, size): return buildPerfectBT_helper(0, size - 1, pre)# Function to print the Inorder of# the given Treedef 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 Callroot = buildPerfectBT(pre, N)# Print Inorder Traversalprint("Inorder traversal of the tree: ", end = "")printInorder(root)# This code is contributed by decode2207. |
C#
// C# program for the above approachusing 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> |
Inorder traversal of the tree: 4 2 5 1 6 3 7
Time Complexity: O(N)
Auxiliary Space: O(N)
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