Linked complete binary tree & its creation

Last Updated : 09 Mar, 2023

A complete binary tree is a binary tree where each level 'l' except the last has 2^l nodes and the nodes at the last level are all left-aligned. Complete binary trees are mainly used in heap-based data structures.
The nodes in the complete binary tree are inserted from left to right in one level at a time. If a level is full, the node is inserted in a new level.
Below are some complete binary trees.

Below binary trees are not complete:

Complete binary trees are generally represented using arrays. The array representation is better because it doesn't contain any empty slots. Given parent index i, its left child is given by 2 * i + 1, and its right child is given by 2 * i + 2. So no extra space is wasted and space to store left and right pointers is saved. However, it may be an interesting programming question to create a Complete Binary Tree using linked representation. Here Linked means a non-array representation where the left and right pointers(or references) are used to refer left and right children respectively. How to write an insert function that always adds a new node in the last level and at the leftmost available position?
To create a linked complete binary tree, we need to keep track of the nodes in a level order fashion such that the next node to be inserted lies in the leftmost position. A queue data structure can be used to keep track of the inserted nodes.

The following are steps to insert a new node in Complete Binary Tree.

If the tree is empty, initialize the root with a new node.
Else, get the front node of the queue.
.......If the left child of this front node doesn't exist, set the left child as the new node.
.......else if the right child of this front node doesn't exist, set the right child as the new node.
If the front node has both the left child and right child, Dequeue() it.
Enqueue() the new node.

Below is the implementation:

C++

// Program for linked implementation of complete binary tree 
#include <bits/stdc++.h>
using namespace std;

// For Queue Size 
#define SIZE 50 

// A tree node 
class node 
{ 
    public:
    int data; 
    node *right,*left; 
}; 

// A queue node 
class Queue 
{ 
    public:
    int front, rear; 
    int size; 
    node**array; 
}; 

// A 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 create a new Queue 
Queue* createQueue(int size) 
{ 
    Queue* queue = new Queue(); 

    queue->front = queue->rear = -1; 
    queue->size = size; 

    queue->array = new node*[queue->size * sizeof( node* )]; 

    int i; 
    for (i = 0; i < size; ++i) 
        queue->array[i] = NULL; 

    return queue; 
} 

// Standard Queue Functions 
int isEmpty(Queue* queue) 
{ 
    return queue->front == -1; 
} 

int isFull(Queue* queue) 
{ return queue->rear == queue->size - 1; } 

int hasOnlyOneItem(Queue* queue) 
{ return queue->front == queue->rear; } 

void Enqueue(node *root, Queue* queue) 
{ 
    if (isFull(queue)) 
        return; 

    queue->array[++queue->rear] = root; 

    if (isEmpty(queue)) 
        ++queue->front; 
} 

node* Dequeue(Queue* queue) 
{ 
    if (isEmpty(queue)) 
        return NULL; 

    node* temp = queue->array[queue->front]; 

    if (hasOnlyOneItem(queue)) 
        queue->front = queue->rear = -1; 
    else
        ++queue->front; 

    return temp; 
} 

node* getFront(Queue* queue) 
{ return queue->array[queue->front]; } 

// A utility function to check if a tree node
// has both left and right children 
int hasBothChild(node* temp) 
{ 
    return temp && temp->left && temp->right; 
} 

// Function to insert a new node in complete binary tree 
void insert(node **root, int data, Queue* queue) 
{ 
    // Create a new node for given data 
    node *temp = newNode(data); 

    // If the tree is empty, initialize the root with new node. 
    if (!*root) 
        *root = temp; 

    else
    { 
        // get the front node of the queue. 
        node* front = getFront(queue); 

        // If the left child of this front node doesn’t exist, set the 
        // left child as the new node 
        if (!front->left) 
            front->left = temp; 

        // If the right child of this front node doesn’t exist, set the 
        // right child as the new node 
        else if (!front->right) 
            front->right = temp; 

        // If the front node has both the left child and right child, 
        // Dequeue() it. 
        if (hasBothChild(front)) 
            Dequeue(queue); 
    } 

    // Enqueue() the new node for later insertions 
    Enqueue(temp, queue); 
} 

// Standard level order traversal to test above function 
void levelOrder(node* root) 
{ 
    Queue* queue = createQueue(SIZE); 

    Enqueue(root, queue); 

    while (!isEmpty(queue)) 
    { 
        node* temp = Dequeue(queue); 

        cout<<temp->data<<" "; 

        if (temp->left) 
            Enqueue(temp->left, queue); 

        if (temp->right) 
            Enqueue(temp->right, queue); 
    } 
} 

// Driver program to test above functions 
int main() 
{ 
    node* root = NULL; 
    Queue* queue = createQueue(SIZE); 
    int i; 

    for(i = 1; i <= 12; ++i) 
        insert(&root, i, queue); 

    levelOrder(root); 

    return 0; 
} 

//This code is contributed by rathbhupendra

// Program for linked implementation of complete binary tree
#include <stdio.h>
#include <stdlib.h>

// For Queue Size
#define SIZE 50

// A tree node
struct node
{
    int data;
    struct node *right,*left;
};

// A queue node
struct Queue
{
    int front, rear;
    int size;
    struct node* *array;
};

// A utility function to create a new tree node
struct node* newNode(int data)
{
    struct node* temp = (struct node*) malloc(sizeof( struct node ));
    temp->data = data;
    temp->left = temp->right = NULL;
    return temp;
}

// A utility function to create a new Queue
struct Queue* createQueue(int size)
{
    struct Queue* queue = (struct Queue*) malloc(sizeof( struct Queue ));

    queue->front = queue->rear = -1;
    queue->size = size;

    queue->array = (struct node**) malloc
                   (queue->size * sizeof( struct node* ));

    int i;
    for (i = 0; i < size; ++i)
        queue->array[i] = NULL;

    return queue;
}

// Standard Queue Functions
int isEmpty(struct Queue* queue)
{
    return queue->front == -1;
}

int isFull(struct Queue* queue)
{  return queue->rear == queue->size - 1; }

int hasOnlyOneItem(struct Queue* queue)
{  return queue->front == queue->rear;  }

void Enqueue(struct node *root, struct Queue* queue)
{
    if (isFull(queue))
        return;

    queue->array[++queue->rear] = root;

    if (isEmpty(queue))
        ++queue->front;
}

struct node* Dequeue(struct Queue* queue)
{
    if (isEmpty(queue))
        return NULL;

    struct node* temp = queue->array[queue->front];

    if (hasOnlyOneItem(queue))
        queue->front = queue->rear = -1;
    else
        ++queue->front;

    return temp;
}

struct node* getFront(struct Queue* queue)
{  return queue->array[queue->front]; }

// A utility function to check if a tree node
// has both left and right children
int hasBothChild(struct node* temp)
{
    return temp && temp->left && temp->right;
}

// Function to insert a new node in complete binary tree
void insert(struct node **root, int data, struct Queue* queue)
{
    // Create a new node for given data
    struct node *temp = newNode(data);

    // If the tree is empty, initialize the root with new node.
    if (!*root)
        *root = temp;

    else
    {
        // get the front node of the queue.
        struct node* front = getFront(queue);

        // If the left child of this front node doesn’t exist, set the
        // left child as the new node
        if (!front->left)
            front->left = temp;

        // If the right child of this front node doesn’t exist, set the
        // right child as the new node
        else if (!front->right)
            front->right = temp;

        // If the front node has both the left child and right child,
        // Dequeue() it.
        if (hasBothChild(front))
            Dequeue(queue);
    }

    // Enqueue() the new node for later insertions
    Enqueue(temp, queue);
}

// Standard level order traversal to test above function
void levelOrder(struct node* root)
{
    struct Queue* queue = createQueue(SIZE);

    Enqueue(root, queue);

    while (!isEmpty(queue))
    {
        struct node* temp = Dequeue(queue);

        printf("%d ", temp->data);

        if (temp->left)
            Enqueue(temp->left, queue);

        if (temp->right)
            Enqueue(temp->right, queue);
    }
}

// Driver program to test above functions
int main()
{
    struct node* root = NULL;
    struct Queue* queue = createQueue(SIZE);
    int i;

    for(i = 1; i <= 12; ++i)
        insert(&root, i, queue);

    levelOrder(root);

    return 0;
}

Java

// Java code for the above approach
import java.util.LinkedList;
import java.util.Queue;

class Node {
  int data;
  Node left, right;
  public Node(int data) {
    this.data = data;
    left = right = null;
  }
}

public class CompleteBinaryTree {
  Node root;

  public CompleteBinaryTree() {
    root = null;
  }

  // A utility function to create a new tree node
  Node newNode(int data) {
    Node temp = new Node(data);
    return temp;
  }

  // Function to insert a new node in complete binary tree
  void insert(int data) {
    // Create a new node for given data
    Node temp = newNode(data);

    // If the tree is empty, initialize the root with new node.
    if (root == null) {
      root = temp;
      return;
    }

    // Create a queue to do level order traversal
    Queue<Node> q = new LinkedList<>();
    q.add(root);

    // Do level order traversal
    while (!q.isEmpty()) {
      Node front = q.peek();

      // If the left child of this front node doesn't exist, set the
      // left child as the new node
      if (front.left == null) {
        front.left = temp;
        break;
      }

      // If the right child of this front node doesn't exist, set the
      // right child as the new node
      else if (front.right == null) {
        front.right = temp;
        break;
      }

      // If the front node has both the left child and right child,
      // remove it from the queue
      else {
        q.remove();
      }

      // Enqueue the left and right children of the current node
      if (front.left != null) {
        q.add(front.left);
      }
      if (front.right != null) {
        q.add(front.right);
      }
    }
  }

  // Standard level order traversal to test above function
  void levelOrder() {
    if (root == null) {
      return;
    }

    Queue<Node> q = new LinkedList<>();
    q.add(root);

    while (!q.isEmpty()) {
      Node temp = q.poll();
      System.out.print(temp.data + " ");

      if (temp.left != null) {
        q.add(temp.left);
      }
      if (temp.right != null) {
        q.add(temp.right);
      }
    }
  }

  public static void main(String[] args) {
    CompleteBinaryTree tree = new CompleteBinaryTree();
    for (int i = 1; i <= 12; i++) {
      tree.insert(i);
    }

    tree.levelOrder();
  }
}

// This code is contributed by ik_7

Python3

# Program for linked implementation 
# of complete binary tree 

# For Queue Size 
SIZE = 50 

# A tree node 
class node:
    
    def __init__(self, data):
      
        self.data = data
        self.right = None
        self.left = None

# A queue node 
class Queue:
    
    def __init__(self):
      
        self.front = None
        self.rear = None
        self.size = 0
        self.array = []

# A utility function to 
# create a new tree node 
def newNode(data):
    
    temp = node(data)
    return temp

# A utility function to 
# create a new Queue 
def createQueue(size):
    
    global queue    
    queue = Queue();
    queue.front = queue.rear = -1; 
    queue.size = size; 
    queue.array = [None for i in range(size)]
    return queue; 
    
# Standard Queue Functions 
def isEmpty(queue):

    return queue.front == -1 

def isFull(queue):
    
    return queue.rear == queue.size - 1; 

def hasOnlyOneItem(queue):
    
    return queue.front == queue.rear; 

def Enqueue(root):

    if (isFull(queue)):
        return; 
    
    queue.rear+=1
    queue.array[queue.rear] = root; 

    if (isEmpty(queue)):
        queue.front+=1; 

def Dequeue():

    if (isEmpty(queue)):
        return None; 

    temp = queue.array[queue.front]; 

    if(hasOnlyOneItem(queue)):
        queue.front = queue.rear = -1; 
    else:
        queue.front+=1 

    return temp; 

def getFront(queue):
    
    return queue.array[queue.front];

# A utility function to check 
# if a tree node has both left 
# and right children 
def hasBothChild(temp):

    return (temp and temp.left and 
            temp.right); 
    
# Function to insert a new 
# node in complete binary tree 
def insert(root, data, queue):

    # Create a new node for
    # given data 
    temp = newNode(data); 

    # If the tree is empty, 
    # initialize the root 
    # with new node. 
    if not root:
        root = temp;
    else:
    
        # get the front node of 
        # the queue. 
        front = getFront(queue); 

        # If the left child of this 
        # front node doesn’t exist,
        # set the left child as the 
        # new node 
        if (not front.left):
            front.left = temp; 

        # If the right child of this 
        # front node doesn’t exist, set 
        # the right child as the new node 
        elif (not front.right):
            front.right = temp; 

        # If the front node has both the 
        # left child and right child, 
        # Dequeue() it. 
        if (hasBothChild(front)): 
            Dequeue();

    # Enqueue() the new node for 
    # later insertions 
    Enqueue(temp); 
    return root
 
# Standard level order 
# traversal to test above
# function 
def levelOrder(root):

    queue = createQueue(SIZE); 
    Enqueue(root); 
    
    while (not isEmpty(queue)):    
        temp = Dequeue();         
        print(temp.data, end = ' ')
        if (temp.left): 
            Enqueue(temp.left); 
        if (temp.right): 
            Enqueue(temp.right); 

# Driver code  
if __name__ == "__main__":
    
    root = None
    queue = createQueue(SIZE); 
    
    for i in range(1, 13):
        root=insert(root, i, 
                    queue); 
     
    levelOrder(root); 

# This code is contributed by Rutvik_56

using System;
using System.Collections.Generic;

class Node
{
    public int data;
    public Node left, right;

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

class CompleteBinaryTree
{
    Node root;

    public CompleteBinaryTree()
    {
        root = null;
    }

    // A utility function to create a new tree node
    Node newNode(int data)
    {
        Node temp = new Node(data);
        return temp;
    }

    // Function to insert a new node in complete binary tree
    void insert(int data)
    {
        // Create a new node for given data
        Node temp = newNode(data);

        // If the tree is empty, initialize the root with new node.
        if (root == null)
        {
            root = temp;
            return;
        }

        // Create a queue to do level order traversal
        Queue<Node> q = new Queue<Node>();
        q.Enqueue(root);

        // Do level order traversal
        while (q.Count > 0)
        {
            Node front = q.Peek();

            // If the left child of this front node doesn't exist, set the
            // left child as the new node
            if (front.left == null)
            {
                front.left = temp;
                break;
            }

            // If the right child of this front node doesn't exist, set the
            // right child as the new node
            else if (front.right == null)
            {
                front.right = temp;
                break;
            }

            // If the front node has both the left child and right child,
            // remove it from the queue
            else
            {
                q.Dequeue();
            }

            // Enqueue the left and right children of the current node
            if (front.left != null)
            {
                q.Enqueue(front.left);
            }
            if (front.right != null)
            {
                q.Enqueue(front.right);
            }
        }
    }

    // Standard level order traversal to test above function
    void levelOrder()
    {
        if (root == null)
        {
            return;
        }

        Queue<Node> q = new Queue<Node>();
        q.Enqueue(root);

        while (q.Count > 0)
        {
            Node temp = q.Dequeue();
            Console.Write(temp.data + " ");

            if (temp.left != null)
            {
                q.Enqueue(temp.left);
            }
            if (temp.right != null)
            {
                q.Enqueue(temp.right);
            }
        }
    }
    
    // Driver program to test above functions
    static void Main(string[] args)
    {
        CompleteBinaryTree tree = new CompleteBinaryTree();
        for (int i = 1; i <= 12; i++)
        {
            tree.insert(i);
        }

        tree.levelOrder();
    }
}

// This code is contributed by Vaibhav.

JavaScript

<script>
// Program for linked implementation 
// of complete binary tree 

// For Queue Size 
const SIZE = 50; 

// A tree node 
class Node {
  constructor(data) {
    this.data = data;
    this.left = null;
    this.right = null;
  }
}

class Queue {
  constructor(size) {
    this.front = -1;
    this.rear = -1;
    this.size = size;
    this.array = new Array(size);
  }

  // Standard Queue Functions 
  isEmpty() {
    return this.front === -1;
  }

  isFull() {
    return this.rear === this.size - 1;
  }

  hasOnlyOneItem() {
    return this.front === this.rear;
  }

  enqueue(root) {
    if (this.isFull()) {
      return;
    }
    this.rear++;
    this.array[this.rear] = root;
    if (this.isEmpty()) {
      this.front++;
    }
  }

  dequeue() {
    if (this.isEmpty()) {
      return null;
    }
    let temp = this.array[this.front];
    if (this.hasOnlyOneItem()) {
      this.front = this.rear = -1;
    } else {
      this.front++;
    }
    return temp;
  }

  getFront() {
    return this.array[this.front];
  }
}

// A utility function to create a new tree node 
function newNode(data) {
  return new Node(data);
}

// A utility function to create a new Queue 
function createQueue(size) {
  let queue = new Queue(size);
  return queue;
}

// A utility function to check if a tree node has both left and right children 
function hasBothChild(temp) {
  return temp && temp.left && temp.right;
}

// Function to insert a new node in complete binary tree 
function insert(root, data, queue) {
  let temp = newNode(data);
  // If the tree is empty, initialize the root with new node. 
  if (!root) {
    root = temp;
  } else {
    // get the front node of the queue. 
    let front = queue.getFront();
    if (!front.left) {
      front.left = temp;
    } else if (!front.right) {
      front.right = temp;
    }
    // If the front node has both the left child and right child, Dequeue() it.
    if (hasBothChild(front)) {
      queue.dequeue();
    }
  }
  // Enqueue() the new node for later insertions 
  queue.enqueue(temp);
  return root;
}

// Standard level order traversal to test above function 

function levelOrder(root) {
  let queue = createQueue(50);
  queue.enqueue(root);
  while (!queue.isEmpty()) {
    let temp = queue.dequeue();
    document.write(temp.data);
    if (temp.left) {
      queue.enqueue(temp.left);
    }
    if (temp.right) {
      queue.enqueue(temp.right);
    }
  }
}

let root = null;
let queue = createQueue(50);

for (let i = 1; i < 13; i++) {
  root = insert(root, i, queue);
}

levelOrder(root);


</script>