Implementing a BST where every node stores the maximum number of nodes in the path till any leaf

Given an array of values. The task is to implement a Binary Search Tree using values of the array where every node stores the maximum number of nodes in the path starting from the node itself and ending at any leaf of the tree.

Note: The maximum number of nodes in the path from any node to any leaf node in a BST is the height of the subtree rooted at that node.

Examples:

Input : arr[] = {1, 2, 3, 4, 5, 6, 7} 
Output :
data = 1 height = 6 
data = 2 height = 5 
data = 3 height = 4 
data = 4 height = 3 
data = 5 height = 2 
data = 6 height = 1 
data = 7 height = 0

Input : arr[] = {4, 12, 10, 5, 11, 8, 7, 6, 9} 
Output :
data = 4 height = 6 
data = 5 height = 3 
data = 6 height = 0 
data = 7 height = 1 
data = 8 height = 2 
data = 9 height = 0 
data = 10 height = 4 
data = 11 height = 0 
data = 12 height = 5 

The idea is to add nodes in the BST fashion. Height of the parent say P will be updated only when the new node is added to the subtree which contributes to the height of P AND (logical) the height of the subtree has increased as well after the addition of the new node.

Let’s say that an existing tree is (data of node is in red and current height of node in green):

Now we are going to add a new node of containing the value 6, the route taken by node in order to get added has been highlighted in blue:

With addition of the new node, the height of it’s immediate parent will be increased (only if the height of the immediate parent of node containing 6 is being affected by this addition – which in this case is true). Once the height of parent is incremented, it will check whether the sub-tree where parent is present is the main contributory to the height of node having that sub-tree as a child, if yes then height of that node will be increased – in short the height incrementation if propagated upwards.

Now we are going to add another node containing value 9 & the path it will take to get added to its final position is in blue:

Since the height of the immediate parent of the node containing value 9 is not getting affected by this addition, its parent’s height won’t get affected and height incrementation won’t get propagated upwards.

Below is the implementation of the above approach: 

C++

// C++ implementation of above approach
#include<bits/stdc++.h>
using namespace std;

// Structure containing basic template od a node
struct node {

    // Stores the data and current height of the node
    int data;
    int height;
    struct node* right;
    struct node* left;
};

int indicator = 0;
void left_insert(struct node*, struct node*);
void right_insert(struct node*, struct node*);

// Inorder traversal of the tree
void traverse(struct node* head)
{
    if (head != NULL) {
        traverse(head->left);
        cout << " data   = " << head->data;
        cout<< " height = " << head->height << endl;
        traverse(head->right);
    }
}

// Insertion to the left sub-tree
void left_insert(struct node* head, struct node* temp)
{
    // Child node of Current head
    struct node* child = NULL;

    if (head->data > temp->data) {
        if (head->left == NULL) {
            indicator = 1;
            child = head->left = temp;
        }
        else {
            left_insert(head->left, temp);
            child = head->left;
        }
    }
    else {
        right_insert(head, temp);
    }

    if ((indicator == 1) && (child != NULL)) {
        if (head->height > child->height) {
            // Ending propagation to height of above nodes
            indicator = 0;
        }
        else {
            head->height += 1;
        }
    }
}

// Insertion to the right sub-tree
void right_insert(struct node* head, struct node* temp)
{
    // Child node of Current head
    struct node* child = NULL;

    if (head->data < temp->data) {
        if (head->right == NULL) {
            indicator = 1;
            child = head->right = temp;
        }
        else {
            right_insert(head->right, temp);
            child = head->right;
        }
    }
    else {
        left_insert(head, temp);
    }

    if ((indicator == 1) && (child != NULL)) {
        if (head->height > child->height) {

            // Ending propagation to height of above nodes
            indicator = 0;
        }
        else {
            head->height += 1;
        }
    }
}

// Function to create node and push
// it to its appropriate position
void add_nodes(struct node** head, int value)
{
    struct node *temp_head = *head, *temp;

    if (*head == NULL) {
        // When first node is added
        *head = new node();
        (*head)->data = value;
        (*head)->height = 0;
        (*head)->right = (*head)->left = NULL;
    }
    else {
        temp = new node();
        temp->data = value;
        temp->height = 0;
        temp->right = temp->left = NULL;
        left_insert(temp_head, temp);
        temp_head = *head;
        indicator = 0;
    }
}

// Driver Code
int main()
{
    struct node *head = NULL, *temp_head = NULL;

    add_nodes(&head, 4);
    add_nodes(&head, 12);
    add_nodes(&head, 10);
    add_nodes(&head, 5);
    add_nodes(&head, 11);
    add_nodes(&head, 8);
    add_nodes(&head, 7);
    add_nodes(&head, 6);
    add_nodes(&head, 9);

    temp_head = head;

    // Traversing the tree to display
    // the updated height values
    traverse(temp_head);
    return 0;
}

// This code is contributed by rrrtnx.

C

// C implementation of above approach
#include <stdio.h>
#include <stdlib.h>

// Structure containing basic template od a node
struct node {

    // Stores the data and current height of the node
    int data;
    int height;
    struct node* right;
    struct node* left;
};

int indicator = 0;
void left_insert(struct node*, struct node*);
void right_insert(struct node*, struct node*);

// Inorder traversal of the tree
void traverse(struct node* head)
{
    if (head != NULL) {
        traverse(head->left);
        printf(" data   = %d", head->data);
        printf(" height = %d\n", head->height);
        traverse(head->right);
    }
}

// Insertion to the left sub-tree
void left_insert(struct node* head, struct node* temp)
{
    // Child node of Current head
    struct node* child = NULL;

    if (head->data > temp->data) {
        if (head->left == NULL) {
            indicator = 1;
            child = head->left = temp;
        }
        else {
            left_insert(head->left, temp);
            child = head->left;
        }
    }
    else {
        right_insert(head, temp);
    }

    if ((indicator == 1) && (child != NULL)) {
        if (head->height > child->height) {
            // Ending propagation to height of above nodes
            indicator = 0;
        }
        else {
            head->height += 1;
        }
    }
}

// Insertion to the right sub-tree
void right_insert(struct node* head, struct node* temp)
{
    // Child node of Current head
    struct node* child = NULL;

    if (head->data < temp->data) {
        if (head->right == NULL) {
            indicator = 1;
            child = head->right = temp;
        }
        else {
            right_insert(head->right, temp);
            child = head->right;
        }
    }
    else {
        left_insert(head, temp);
    }

    if ((indicator == 1) && (child != NULL)) {
        if (head->height > child->height) {

            // Ending propagation to height of above nodes
            indicator = 0;
        }
        else {
            head->height += 1;
        }
    }
}

// Function to create node and push
// it to its appropriate position
void add_nodes(struct node** head, int value)
{
    struct node *temp_head = *head, *temp;

    if (*head == NULL) {
        // When first node is added
        *head = malloc(sizeof(**head));
        (*head)->data = value;
        (*head)->height = 0;
        (*head)->right = (*head)->left = NULL;
    }
    else {
        temp = malloc(sizeof(*temp));
        temp->data = value;
        temp->height = 0;
        temp->right = temp->left = NULL;
        left_insert(temp_head, temp);
        temp_head = *head;
        indicator = 0;
    }
}

// Driver Code
int main()
{
    struct node *head = NULL, *temp_head = NULL;

    add_nodes(&head, 4);
    add_nodes(&head, 12);
    add_nodes(&head, 10);
    add_nodes(&head, 5);
    add_nodes(&head, 11);
    add_nodes(&head, 8);
    add_nodes(&head, 7);
    add_nodes(&head, 6);
    add_nodes(&head, 9);

    temp_head = head;

    // Traversing the tree to display
    // the updated height values
    traverse(temp_head);
    return 0;
}

Java

// Java implementation of above approach
class GFG
{

// Structure containing basic template od a node
static class node
{

    // Stores the data and current height of the node
    int data;
    int height;
    node right;
    node left;
}

static int indicator = 0;

// Inorder traversal of the tree
static void traverse(node head)
{
    if (head != null)
    {
        traverse(head.left);
        System.out.printf(" data = %d", head.data);
        System.out.printf(" height = %d\n", head.height);
        traverse(head.right);
    }
}

// Insertion to the left sub-tree
static void left_insert(node head, node temp)
{
    // Child node of Current head
    node child = null;

    if (head.data > temp.data)
    {
        if (head.left == null)
        {
            indicator = 1;
            child = head.left = temp;
        }
        else
        {
            left_insert(head.left, temp);
            child = head.left;
        }
    }
    else
    {
        right_insert(head, temp);
    }

    if ((indicator == 1) && (child != null))
    {
        if (head.height > child.height)
        {

            // Ending propagation to height of above nodes
            indicator = 0;
        }
        else
        {
            head.height += 1;
        }
    }
}

// Insertion to the right sub-tree
static void right_insert(node head, node temp)
{
    // Child node of Current head
    node child = null;

    if (head.data < temp.data)
    {
        if (head.right == null)
        {
            indicator = 1;
            child = head.right = temp;
        }
        else
        {
            right_insert(head.right, temp);
            child = head.right;
        }
    }
    else
    {
        left_insert(head, temp);
    }

    if ((indicator == 1) && (child != null))
    {
        if (head.height > child.height)
        {

            // Ending propagation to height of above nodes
            indicator = 0;
        }
        else
        {
            head.height += 1;
        }
    }
}

// Function to create node and push
// it to its appropriate position
static node add_nodes(node head, int value)
{
    node temp_head = head, temp;

    if (head == null)
    {
        // When first node is added
        head = new node();
        (head).data = value;
        (head).height = 0;
        (head).right = (head).left = null;
    }
    else
    {
        temp = new node();
        temp.data = value;
        temp.height = 0;
        temp.right = temp.left = null;
        left_insert(temp_head, temp);
        temp_head = head;
        indicator = 0;
    }
    return head;
}

// Driver Code
public static void main(String args[])
{
    node head = null, temp_head = null;

    head = add_nodes(head, 4);
    head = add_nodes(head, 12);
    head = add_nodes(head, 10);
    head = add_nodes(head, 5);
    head = add_nodes(head, 11);
    head = add_nodes(head, 8);
    head = add_nodes(head, 7);
    head = add_nodes(head, 6);
    head = add_nodes(head, 9);

    temp_head = head;

    // Traversing the tree to display
    // the updated height values
    traverse(temp_head);
}
}

// This code is contributed by Arnab Kundu

Python3

# Python implementation of above approach

# Structure containing basic template od a node
class node:
    def __init__(self) -> None:

        # Stores the data and current height of the node
        self.data = 0
        self.height = 0
        self.right = None
        self.left = None

# Inorder traversal of the tree
def traverse(head: node) -> None:

    if (head != None):
        traverse(head.left)
        print(" data = {}".format(head.data), end="")
        print(" height = {}".format(head.height))
        traverse(head.right)

# Insertion to the left sub-tree
def left_insert(head: node, temp: node) -> None:
    global indicator

    # Child node of Current head
    child = None
    if (head.data > temp.data):
        if (head.left == None):
            indicator = 1
            child = head.left = temp
        else:
            left_insert(head.left, temp)
            child = head.left
    else:
        right_insert(head, temp)

    if ((indicator == 1) and (child != None)):
        if (head.height > child.height):

            # Ending propagation to height of above nodes
            indicator = 0

        else:
            head.height += 1

# Insertion to the right sub-tree
def right_insert(head: node, temp: node) -> None:
    global indicator

    # Child node of Current head
    child = None

    if (head.data < temp.data):
        if (head.right == None):
            indicator = 1
            child = head.right = temp

        else:
            right_insert(head.right, temp)
            child = head.right

    else:
        left_insert(head, temp)

    if ((indicator == 1) and (child != None)):
        if (head.height > child.height):

            # Ending propagation to height of above nodes
            indicator = 0

        else:
            head.height += 1

# Function to create node and push
# it to its appropriate position
def add_nodes(head: node, value: int) -> node:
    global indicator
    temp_head = head
    temp = None

    if (head == None):
        # When first node is added
        head = node()
        (head).data = value
        (head).height = 0
        (head).right = (head).left = None

    else:
        temp = node()
        temp.data = value
        temp.height = 0
        temp.right = temp.left = None
        left_insert(temp_head, temp)
        temp_head = head
        indicator = 0
    return head

# Driver Code
if __name__ == "__main__":
    indicator = 0
    head = None
    temp_head = None

    head = add_nodes(head, 4)
    head = add_nodes(head, 12)
    head = add_nodes(head, 10)
    head = add_nodes(head, 5)
    head = add_nodes(head, 11)
    head = add_nodes(head, 8)
    head = add_nodes(head, 7)
    head = add_nodes(head, 6)
    head = add_nodes(head, 9)

    temp_head = head

    # Traversing the tree to display
    # the updated height values
    traverse(temp_head)

# This code is contributed by sanjeev2552

C#

// C# implementation of above approach
using System;

public class GFG {

    // Structure containing basic template od a node
    public class node {

        // Stores the data and current height of the node
        public int data;
        public int height;
        public node right;
        public node left;
    }

    static int indicator = 0;

    // Inorder traversal of the tree
    static void traverse(node head) {
        if (head != null) {
            traverse(head.left);
            Console.Write(" data = {0}", head.data);
            Console.Write(" height = {0}\n", head.height);
            traverse(head.right);
        }
    }

    // Insertion to the left sub-tree
    static void left_insert(node head, node temp) {
        // Child node of Current head
        node child = null;

        if (head.data > temp.data) {
            if (head.left == null) {
                indicator = 1;
                child = head.left = temp;
            } else {
                left_insert(head.left, temp);
                child = head.left;
            }
        } else {
            right_insert(head, temp);
        }

        if ((indicator == 1) && (child != null)) {
            if (head.height > child.height) {

                // Ending propagation to height of above nodes
                indicator = 0;
            } else {
                head.height += 1;
            }
        }
    }

    // Insertion to the right sub-tree
    static void right_insert(node head, node temp) {
        // Child node of Current head
        node child = null;

        if (head.data < temp.data) {
            if (head.right == null) {
                indicator = 1;
                child = head.right = temp;
            } else {
                right_insert(head.right, temp);
                child = head.right;
            }
        } else {
            left_insert(head, temp);
        }

        if ((indicator == 1) && (child != null)) {
            if (head.height > child.height) {

                // Ending propagation to height of above nodes
                indicator = 0;
            } else {
                head.height += 1;
            }
        }
    }

    // Function to create node and push
    // it to its appropriate position
    static node add_nodes(node head, int value) {
        node temp_head = head, temp;

        if (head == null) {
            // When first node is added
            head = new node();
            (head).data = value;
            (head).height = 0;
            (head).right = (head).left = null;
        } else {
            temp = new node();
            temp.data = value;
            temp.height = 0;
            temp.right = temp.left = null;
            left_insert(temp_head, temp);
            temp_head = head;
            indicator = 0;
        }
        return head;
    }

    // Driver Code
    public static void Main(String []args) {
        node head = null, temp_head = null;

        head = add_nodes(head, 4);
        head = add_nodes(head, 12);
        head = add_nodes(head, 10);
        head = add_nodes(head, 5);
        head = add_nodes(head, 11);
        head = add_nodes(head, 8);
        head = add_nodes(head, 7);
        head = add_nodes(head, 6);
        head = add_nodes(head, 9);

        temp_head = head;

        // Traversing the tree to display
        // the updated height values
        traverse(temp_head);
    }
}

// This code contributed by Rajput-Ji

Javascript

<script>

// Javascript implementation of above approach

// Structure containing basic template od a node
class node
{
    constructor()
    {
        this.left;
        this.right;
        this.data;
        this.height;
    }
}

let indicator = 0;

// Inorder traversal of the tree
function traverse(head)
{
    if (head != null)
    {
        traverse(head.left);
        document.write(" data = " + head.data);
        document.write(" height = "+
                       head.height + "</br>");
        traverse(head.right);
    }
}

// Insertion to the left sub-tree
function left_insert(head, temp)
{

    // Child node of Current head
    let child = null;

    if (head.data > temp.data)
    {
        if (head.left == null)
        {
            indicator = 1;
            child = head.left = temp;
        }
        else
        {
            left_insert(head.left, temp);
            child = head.left;
        }
    }
    else
    {
        right_insert(head, temp);
    }

    if ((indicator == 1) && (child != null))
    {
        if (head.height > child.height)
        {

            // Ending propagation to height
            // of above nodes
            indicator = 0;
        }
        else
        {
            head.height += 1;
        }
    }
}

// Insertion to the right sub-tree
function right_insert(head, temp)
{

    // Child node of Current head
    let child = null;

    if (head.data < temp.data)
    {
        if (head.right == null)
        {
            indicator = 1;
            child = head.right = temp;
        }
        else
        {
            right_insert(head.right, temp);
            child = head.right;
        }
    }
    else
    {
        left_insert(head, temp);
    }

    if ((indicator == 1) && (child != null))
    {
        if (head.height > child.height)
        {

            // Ending propagation to height
            // of above nodes
            indicator = 0;
        }
        else
        {
            head.height += 1;
        }
    }
}

// Function to create node and push
// it to its appropriate position
function add_nodes(head, value)
{
    let temp_head = head, temp;

    if (head == null)
    {

        // When first node is added
        head = new node();
        (head).data = value;
        (head).height = 0;
        (head).right = (head).left = null;
    }
    else
    {
        temp = new node();
        temp.data = value;
        temp.height = 0;
        temp.right = temp.left = null;
        left_insert(temp_head, temp);
        temp_head = head;
        indicator = 0;
    }
    return head;
}

// Driver code
let head = null, temp_head = null;

head = add_nodes(head, 4);
head = add_nodes(head, 12);
head = add_nodes(head, 10);
head = add_nodes(head, 5);
head = add_nodes(head, 11);
head = add_nodes(head, 8);
head = add_nodes(head, 7);
head = add_nodes(head, 6);
head = add_nodes(head, 9);

temp_head = head;

// Traversing the tree to display
// the updated height values
traverse(temp_head);

// This code is contributed by divyeshrabadiya07

</script>

Output:

 data   = 4 height = 6
 data   = 5 height = 3
 data   = 6 height = 0
 data   = 7 height = 1
 data   = 8 height = 2
 data   = 9 height = 0
 data   = 10 height = 4
 data   = 11 height = 0
 data   = 12 height = 5

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