Maximum element between two nodes of BST

Given an array of N elements and two integers A, B which belong to the given array. Create a Binary Search Tree by inserting elements from arr[0] to arr[n-1]. The task is to find the maximum element in the path from A to B.

Examples :

Input : arr[] = { 18, 36, 9, 6, 12, 10, 1, 8 }, 
        a = 1, 
        b = 10.
        
Output : 12
 

Path from 1 to 10 contains { 1, 6, 9, 12, 10 }. The maximum element is 12.

The idea is to find Lowest Common Ancestor of node ‘a’ and node ‘b’. Then search maximum node between LCA and ‘a’, and also find the maximum node between LCA and ‘b’. The answer will be maximum node of two.

Implementation:

C++

// C++ program to find maximum element in the path
// between two Nodes of Binary Search Tree.
#include <bits/stdc++.h>
using namespace std;

struct Node
{
    struct Node *left, *right;
    int data;
};

// Create and return a pointer of new Node.
Node *createNode(int x)
{
    Node *p = new Node;
    p -> data = x;
    p -> left = p -> right = NULL;
    return p;
}

// Insert a new Node in Binary Search Tree.
void insertNode(struct Node *root, int x)
{
    Node *p = root, *q = NULL;

    while (p != NULL)
    {
        q = p;
        if (p -> data < x)
            p = p -> right;
        else
            p = p -> left;
    }

    if (q == NULL)
        p = createNode(x);
    else
    {
        if (q -> data < x)
            q -> right = createNode(x);
        else
            q -> left = createNode(x);
    }
}

// Return the maximum element between a Node
// and its given ancestor.
int maxelpath(Node *q, int x)
{
    Node *p = q;

    int mx = INT_MIN;

    // Traversing the path between ancestor and
    // Node and finding maximum element.
    while (p -> data != x)
    {
        if (p -> data > x)
        {
            mx = max(mx, p -> data);
            p = p -> left;
        }
        else
        {
            mx = max(mx, p -> data);
            p = p -> right;
        }
    }

    return max(mx, x);
}

// Return maximum element in the path between
// two given Node of BST.
int maximumElement(struct Node *root, int x, int y)
{
    Node *p = root;

    // Finding the LCA of Node x and Node y
    while ((x < p -> data && y < p -> data) ||
        (x > p -> data && y > p -> data))
    {
        // Checking if both the Node lie on the
        // left side of the parent p.
        if (x < p -> data && y < p -> data)
            p = p -> left;

        // Checking if both the Node lie on the
        // right side of the parent p.
        else if (x > p -> data && y > p -> data)
            p = p -> right;
    }

    // Return the maximum of maximum elements occur
    // in path from ancestor to both Node.
    return max(maxelpath(p, x), maxelpath(p, y));
}


// Driver Code
int main()
{
    int arr[] = { 18, 36, 9, 6, 12, 10, 1, 8 };
    int a = 1, b = 10;
    int n = sizeof(arr) / sizeof(arr[0]);

    // Creating the root of Binary Search Tree
    struct Node *root = createNode(arr[0]);

    // Inserting Nodes in Binary Search Tree
    for (int i = 1; i < n; i++)
        insertNode(root, arr[i]);

    cout << maximumElement(root, a, b) << endl;

    return 0;
}

Java

// Java program to find maximum element in the path
// between two Nodes of Binary Search Tree.
class Solution
{

static class Node
{
     Node left, right;
    int data;
}

// Create and return a pointer of new Node.
static Node createNode(int x)
{
    Node p = new Node();
    p . data = x;
    p . left = p . right = null;
    return p;
}

// Insert a new Node in Binary Search Tree.
static void insertNode( Node root, int x)
{
    Node p = root, q = null;

    while (p != null)
    {
        q = p;
        if (p . data < x)
            p = p . right;
        else
            p = p . left;
    }

    if (q == null)
        p = createNode(x);
    else
    {
        if (q . data < x)
            q . right = createNode(x);
        else
            q . left = createNode(x);
    }
}

// Return the maximum element between a Node
// and its given ancestor.
static int maxelpath(Node q, int x)
{
    Node p = q;

    int mx = -1;

    // Traversing the path between ancestor and
    // Node and finding maximum element.
    while (p . data != x)
    {
        if (p . data > x)
        {
            mx = Math.max(mx, p . data);
            p = p . left;
        }
        else
        {
            mx = Math.max(mx, p . data);
            p = p . right;
        }
    }

    return Math.max(mx, x);
}

// Return maximum element in the path between
// two given Node of BST.
static int maximumElement( Node root, int x, int y)
{
    Node p = root;

    // Finding the LCA of Node x and Node y
    while ((x < p . data && y < p . data) ||
        (x > p . data && y > p . data))
    {
        // Checking if both the Node lie on the
        // left side of the parent p.
        if (x < p . data && y < p . data)
            p = p . left;

        // Checking if both the Node lie on the
        // right side of the parent p.
        else if (x > p . data && y > p . data)
            p = p . right;
    }

    // Return the maximum of maximum elements occur
    // in path from ancestor to both Node.
    return Math.max(maxelpath(p, x), maxelpath(p, y));
}


// Driver Code
public static void main(String args[])
{
    int arr[] = { 18, 36, 9, 6, 12, 10, 1, 8 };
    int a = 1, b = 10;
    int n =arr.length;

    // Creating the root of Binary Search Tree
     Node root = createNode(arr[0]);

    // Inserting Nodes in Binary Search Tree
    for (int i = 1; i < n; i++)
        insertNode(root, arr[i]);

    System.out.println( maximumElement(root, a, b) );

}
}
//contributed by Arnab Kundu

Python3

# Python 3 program to find maximum element
# in the path between two Nodes of Binary
# Search Tree.

# Create and return a pointer of new Node.
class createNode:

    # Constructor to create a new node
    def __init__(self, data):
        self.data = data
        self.left = None
        self.right = None

# Insert a new Node in Binary Search Tree.
def insertNode(root, x):
    p, q = root, None

    while p != None:
        q = p
        if p.data < x:
            p = p.right
        else:
            p = p.left

    if q == None:
        p = createNode(x)
    else:
        if q.data < x:
            q.right = createNode(x)
        else:
            q.left = createNode(x)

# Return the maximum element between a
# Node and its given ancestor.
def maxelpath(q, x):
    p = q

    mx = -999999999999

    # Traversing the path between ancestor
    # and Node and finding maximum element.
    while p.data != x:
        if p.data > x:
            mx = max(mx, p.data)
            p = p.left
        else:
            mx = max(mx, p.data)
            p = p.right

    return max(mx, x)

# Return maximum element in the path
# between two given Node of BST.
def maximumElement(root, x, y):
    p = root

    # Finding the LCA of Node x and Node y
    while ((x < p.data and y < p.data) or
           (x > p.data and y > p.data)):

        # Checking if both the Node lie on
        # the left side of the parent p.
        if x < p.data and y < p.data:
            p = p.left

        # Checking if both the Node lie on
        # the right side of the parent p.
        elif x > p.data and y > p.data:
            p = p.right

    # Return the maximum of maximum elements
    # occur in path from ancestor to both Node.
    return max(maxelpath(p, x), maxelpath(p, y))

# Driver Code
if __name__ == '__main__':
    arr = [ 18, 36, 9, 6, 12, 10, 1, 8]
    a, b = 1, 10
    n = len(arr)

    # Creating the root of Binary Search Tree
    root = createNode(arr[0])

    # Inserting Nodes in Binary Search Tree
    for i in range(1,n):
        insertNode(root, arr[i])

    print(maximumElement(root, a, b))

# This code is contributed by PranchalK

C#

using System;

// C# program to find maximum element in the path
// between two Nodes of Binary Search Tree.
public class Solution
{

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

// Create and return a pointer of new Node.
public static Node createNode(int x)
{
    Node p = new Node();
    p.data = x;
    p.left = p.right = null;
    return p;
}

// Insert a new Node in Binary Search Tree.
public static void insertNode(Node root, int x)
{
    Node p = root, q = null;

    while (p != null)
    {
        q = p;
        if (p.data < x)
        {
            p = p.right;
        }
        else
        {
            p = p.left;
        }
    }

    if (q == null)
    {
        p = createNode(x);
    }
    else
    {
        if (q.data < x)
        {
            q.right = createNode(x);
        }
        else
        {
            q.left = createNode(x);
        }
    }
}

// Return the maximum element between a Node
// and its given ancestor.
public static int maxelpath(Node q, int x)
{
    Node p = q;

    int mx = -1;

    // Traversing the path between ancestor and
    // Node and finding maximum element.
    while (p.data != x)
    {
        if (p.data > x)
        {
            mx = Math.Max(mx, p.data);
            p = p.left;
        }
        else
        {
            mx = Math.Max(mx, p.data);
            p = p.right;
        }
    }

    return Math.Max(mx, x);
}

// Return maximum element in the path between
// two given Node of BST.
public static int maximumElement(Node root, int x, int y)
{
    Node p = root;

    // Finding the LCA of Node x and Node y
    while ((x < p.data && y < p.data) || (x > p.data && y > p.data))
    {
        // Checking if both the Node lie on the
        // left side of the parent p.
        if (x < p.data && y < p.data)
        {
            p = p.left;
        }

        // Checking if both the Node lie on the
        // right side of the parent p.
        else if (x > p.data && y > p.data)
        {
            p = p.right;
        }
    }

    // Return the maximum of maximum elements occur
    // in path from ancestor to both Node.
    return Math.Max(maxelpath(p, x), maxelpath(p, y));
}


// Driver Code
public static void Main(string[] args)
{
    int[] arr = new int[] {18, 36, 9, 6, 12, 10, 1, 8};
    int a = 1, b = 10;
    int n = arr.Length;

    // Creating the root of Binary Search Tree
     Node root = createNode(arr[0]);

    // Inserting Nodes in Binary Search Tree
    for (int i = 1; i < n; i++)
    {
        insertNode(root, arr[i]);
    }

    Console.WriteLine(maximumElement(root, a, b));

}
}

  //  This code is contributed by Shrikant13

Javascript

<script>

// JavaScript program to find
// maximum element in the path
// between two Nodes of Binary
// Search Tree.


     class Node {
            constructor(val) {
                this.data = val;
                this.left = null;
                this.right = null;
            }
        }


    // Create and return a pointer of new Node.
    function createNode(x) {
var p = new Node();
        p.data = x;
        p.left = p.right = null;
        return p;
    }

    // Insert a new Node in Binary Search Tree.
    function insertNode(root , x) {
       var p = root, q = null;

        while (p != null) {
            q = p;
            if (p.data < x)
                p = p.right;
            else
                p = p.left;
        }

        if (q == null)
            p = createNode(x);
        else {
            if (q.data < x)
                q.right = createNode(x);
            else
                q.left = createNode(x);
        }
    }

    // Return the maximum element between a Node
    // and its given ancestor.
    function maxelpath(q , x) {
        var p = q;

        var mx = -1;

        // Traversing the path between ancestor and
        // Node and finding maximum element.
        while (p.data != x) {
            if (p.data > x) {
                mx = Math.max(mx, p.data);
                p = p.left;
            } else {
                mx = Math.max(mx, p.data);
                p = p.right;
            }
        }

        return Math.max(mx, x);
    }

    // Return maximum element in the path between
    // two given Node of BST.
    function maximumElement(root , x , y) {
     var p = root;

        // Finding the LCA of Node x and Node y
        while ((x < p.data && y < p.data) ||
        (x > p.data && y > p.data)) {
            // Checking if both the Node lie on the
            // left side of the parent p.
            if (x < p.data && y < p.data)
                p = p.left;

            // Checking if both the Node lie on the
            // right side of the parent p.
            else if (x > p.data && y > p.data)
                p = p.right;
        }

        // Return the maximum of maximum elements occur
        // in path from ancestor to both Node.
        return Math.max(maxelpath(p, x), maxelpath(p, y));
    }

    // Driver Code

        var arr = [ 18, 36, 9, 6, 12, 10, 1, 8 ];
        var a = 1, b = 10;
        var n = arr.length;

        // Creating the root of Binary Search Tree
        var root = createNode(arr[0]);

        // Inserting Nodes in Binary Search Tree
        for (i = 1; i < n; i++)
            insertNode(root, arr[i]);

        document.write(maximumElement(root, a, b));


// This code contributed by gauravrajput1

</script>

Output

12

Time complexity: O(h), where h is the height of BST
Auxiliary Space: O(1)

Approach: Use a hash table to store the parent node of each node in the binary search tree. We can start from both the given nodes and traverse up the tree, storing the nodes encountered in a set. Once we reach the root or a common ancestor of the two nodes, we can traverse down the tree from each node and find the maximum element encountered in the set of nodes.

Algorithm steps for the above approach:

• Create an empty hash table to store the parent node of each node in the binary search tree.
• Perform a depth-first search (DFS) traversal of the binary search tree and populate the hash table with the parent node of each node.
• Initialize two pointers, say p1 and p2, to the given nodes.
• Initialize two empty sets, say s1 and s2, to store the nodes encountered while traversing up the tree from p1 and p2, respectively.
• While p1 and p2 are not equal, do the following:
  a. If p1 is not null, add it to set s1 and update p1 to its parent node using the hash table.
  b. If p2 is not null, add it to set s2 and update p2 to its parent node using the hash table.
• Find the intersection set of s1 and s2, i.e., the set of nodes that are common to both s1 and s2.
• Initialize a variable, say max_element, to the minimum integer value.
• For each node in the intersection set, traverse down the tree from the node to the given nodes and find the maximum element encountered. Update max_element if a greater element is encountered.
• Return max_element.

Below is the implementation of the above approach:

C++

// C++ program to find maximum element in the path
// between two Nodes of Binary Search Tree.
#include <bits/stdc++.h>
using namespace std;

struct Node
{
    struct Node *left, *right;
    int data;
};

// Create and return a pointer of new Node.
Node *createNode(int x)
{
    Node *p = new Node;
    p -> data = x;
    p -> left = p -> right = NULL;
    return p;
}

// Insert a new Node in Binary Search Tree.
void insertNode(struct Node *root, int x)
{
    Node *p = root, *q = NULL;

    while (p != NULL)
    {
        q = p;
        if (p -> data < x)
            p = p -> right;
        else
            p = p -> left;
    }

    if (q == NULL)
        p = createNode(x);
    else
    {
        if (q -> data < x)
            q -> right = createNode(x);
        else
            q -> left = createNode(x);
    }
}

// Return the maximum element between a Node
// and its given ancestor.
int maxelpath(Node *q, int x)
{
    Node *p = q;

    int mx = INT_MIN;

    // Traversing the path between ancestor and
    // Node and finding maximum element.
    while (p -> data != x)
    {
        if (p -> data > x)
        {
            mx = max(mx, p -> data);
            p = p -> left;
        }
        else
        {
            mx = max(mx, p -> data);
            p = p -> right;
        }
    }

    return max(mx, x);
}

// Return maximum element in the path between
// two given Node of BST.
int maximumElement(struct Node *root, int x, int y)
{
    Node *p = root;

    // Finding the LCA of Node x and Node y
    while ((x < p -> data && y < p -> data) ||
        (x > p -> data && y > p -> data))
    {
        // Checking if both the Node lie on the
        // left side of the parent p.
        if (x < p -> data && y < p -> data)
            p = p -> left;

        // Checking if both the Node lie on the
        // right side of the parent p.
        else if (x > p -> data && y > p -> data)
            p = p -> right;
    }

    // Return the maximum of maximum elements occur
    // in path from ancestor to both Node.
    return max(maxelpath(p, x), maxelpath(p, y));
}


// Driver Code
int main()
{
    int arr[] = { 18, 36, 9, 6, 12, 10, 1, 8 };
    int a = 1, b = 10;
    int n = sizeof(arr) / sizeof(arr[0]);

    // Creating the root of Binary Search Tree
    struct Node *root = createNode(arr[0]);

    // Inserting Nodes in Binary Search Tree
    for (int i = 1; i < n; i++)
        insertNode(root, arr[i]);

    cout << maximumElement(root, a, b) << endl;

    return 0;
}

Java

import java.util.*;

class Node {
    Node left, right;
    int data;

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

public class Main {
    // Create and return a pointer to a new Node
    static Node createNode(int x) {
        Node p = new Node(x);
        p.left = p.right = null;
        return p;
    }

    // Insert a new Node in the Binary Search Tree
    static void insertNode(Node root, int x) {
        Node p = root, q = null;

        while (p != null) {
            q = p;
            if (p.data < x)
                p = p.right;
            else
                p = p.left;
        }

        if (q == null)
            p = createNode(x);
        else {
            if (q.data < x)
                q.right = createNode(x);
            else
                q.left = createNode(x);
        }
    }

    // Return the maximum element between a Node
    // and its given ancestor.
    static int maxelpath(Node q, int x) {
        Node p = q;

        int mx = Integer.MIN_VALUE;

        // Traversing the path between ancestor and Node
        // and finding the maximum element.
        while (p.data != x) {
            if (p.data > x) {
                mx = Math.max(mx, p.data);
                p = p.left;
            } else {
                mx = Math.max(mx, p.data);
                p = p.right;
            }
        }

        return Math.max(mx, x);
    }

    // Return maximum element in the path between
    // two given Nodes of BST.
    static int maximumElement(Node root, int x, int y) {
        Node p = root;

        // Finding the LCA (Lowest Common Ancestor) of Node x and Node y
        while ((x < p.data && y < p.data) || (x > p.data && y > p.data)) {
            // Checking if both the Nodes lie on the left side of the parent p.
            if (x < p.data && y < p.data)
                p = p.left;
            // Checking if both the Nodes lie on the right side of the parent p.
            else if (x > p.data && y > p.data)
                p = p.right;
        }

        // Return the maximum of maximum elements occurring
        // in the path from the ancestor to both Nodes.
        return Math.max(maxelpath(p, x), maxelpath(p, y));
    }

    // Driver Code
    public static void main(String[] args) {
        int[] arr = { 18, 36, 9, 6, 12, 10, 1, 8 };
        int a = 1, b = 10;
        int n = arr.length;

        // Creating the root of Binary Search Tree
        Node root = createNode(arr[0]);

        // Inserting Nodes in Binary Search Tree
        for (int i = 1; i < n; i++)
            insertNode(root, arr[i]);

        System.out.println(maximumElement(root, a, b));
    }
}

Python3

# Python program for the above approach
class Node:
    def __init__(self, data):
        self.data = data
        self.left = None
        self.right = None

# Insert a new Node in Binary Search Tree.
def insertNode(root, x):
    if root is None:
        return Node(x)

    if x < root.data:
        root.left = insertNode(root.left, x)
    elif x > root.data:
        root.right = insertNode(root.right, x)

    return root

# Return the maximum element between a Node and its given ancestor.
def maxelpath(q, x):
    p = q
    mx = float('-inf')

    # Traversing the path between ancestor and Node and finding the maximum element.
    while p.data != x:
        if p.data > x:
            mx = max(mx, p.data)
            p = p.left
        else:
            mx = max(mx, p.data)
            p = p.right

    return max(mx, x)

# Return maximum element in the path between two given Node of BST.
def maximumElement(root, x, y):
    p = root

    # Finding the LCA of Node x and Node y
    while (x < p.data and y < p.data) or (x > p.data and y > p.data):
        # Checking if both the Node lie on the left side of the parent p.
        if x < p.data and y < p.data:
            p = p.left

        # Checking if both the Node lie on the right side of the parent p.
        elif x > p.data and y > p.data:
            p = p.right

    # Return the maximum of maximum elements occur in path from ancestor to both Node.
    return max(maxelpath(p, x), maxelpath(p, y))

# Driver Code
arr = [18, 36, 9, 6, 12, 10, 1, 8]
a, b = 1, 10
n = len(arr)

# Creating the root of Binary Search Tree
root = None
for i in range(n):
    root = insertNode(root, arr[i])

print(maximumElement(root, a, b))

C#

using System;

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

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

class Program
{
    // Create and return a pointer of a new Node.
    static Node CreateNode(int x)
    {
        Node p = new Node(x);
        return p;
    }

    // Insert a new Node in Binary Search Tree.
    static void InsertNode(Node root, int x)
    {
        Node p = root;
        Node q = null;

        while (p != null)
        {
            q = p;
            if (p.data < x)
                p = p.right;
            else
                p = p.left;
        }

        if (q == null)
            p = CreateNode(x);
        else
        {
            if (q.data < x)
                q.right = CreateNode(x);
            else
                q.left = CreateNode(x);
        }
    }

    // Return the maximum element between a Node
    // and its given ancestor.
    static int MaxPath(Node q, int x)
    {
        Node p = q;
        int mx = int.MinValue;

        // Traversing the path between ancestor and
        // Node and finding the maximum element.
        while (p.data != x)
        {
            if (p.data > x)
            {
                mx = Math.Max(mx, p.data);
                p = p.left;
            }
            else
            {
                mx = Math.Max(mx, p.data);
                p = p.right;
            }
        }

        return Math.Max(mx, x);
    }

    // Return the maximum element in the path between
    // two given Nodes of BST.
    static int MaximumElement(Node root, int x, int y)
    {
        Node p = root;

        // Finding the LCA of Node x and Node y
        while ((x < p.data && y < p.data) || (x > p.data && y > p.data))
        {
            // Checking if both the Nodes lie on the
            // left side of the parent p.
            if (x < p.data && y < p.data)
                p = p.left;

            // Checking if both the Nodes lie on the
            // right side of the parent p.
            else if (x > p.data && y > p.data)
                p = p.right;
        }

        // Return the maximum of maximum elements occur
        // in the path from ancestor to both Nodes.
        return Math.Max(MaxPath(p, x), MaxPath(p, y));
    }

    // Driver Code
    static void Main()
    {
        int[] arr = { 18, 36, 9, 6, 12, 10, 1, 8 };
        int a = 1, b = 10;
        int n = arr.Length;

        // Creating the root of Binary Search Tree
        Node root = new Node(arr[0]);

        // Inserting Nodes in Binary Search Tree
        for (int i = 1; i < n; i++)
            InsertNode(root, arr[i]);

        Console.WriteLine(MaximumElement(root, a, b));
    }
}

// This code is contrbuted by shivamgupta0987654321

Javascript

// JavaScript Program for the above approach
class Node {
    constructor(data) {
        this.data = data;
        this.left = null;
        this.right = null;
    }
}

// Insert a new Node in Binary Search Tree.
function insertNode(root, x) {
    if (root === null) {
        return new Node(x);
    }

    if (x < root.data) {
        root.left = insertNode(root.left, x);
    } else if (x > root.data) {
        root.right = insertNode(root.right, x);
    }

    return root;
}

// Return the maximum element between a Node and its given ancestor.
function maxelpath(q, x) {
    let p = q;
    let mx = Number.MIN_SAFE_INTEGER;

    // Traversing the path between ancestor and Node and finding the maximum element.
    while (p.data !== x) {
        if (p.data > x) {
            mx = Math.max(mx, p.data);
            p = p.left;
        } else {
            mx = Math.max(mx, p.data);
            p = p.right;
        }
    }

    return Math.max(mx, x);
}

// Return maximum element in the path between two given Node of BST.
function maximumElement(root, x, y) {
    let p = root;

    // Finding the LCA of Node x and Node y
    while ((x < p.data && y < p.data) || (x > p.data && y > p.data)) {
        // Checking if both the Node lie on the left side of the parent p.
        if (x < p.data && y < p.data) {
            p = p.left;
        }

        // Checking if both the Node lie on the right side of the parent p.
        else if (x > p.data && y > p.data) {
            p = p.right;
        }
    }

    // Return the maximum of maximum elements occur in path from ancestor to both Node.
    return Math.max(maxelpath(p, x), maxelpath(p, y));
}

// Driver Code
const arr = [18, 36, 9, 6, 12, 10, 1, 8];
const a = 1, b = 10;
const n = arr.length;

// Creating the root of Binary Search Tree
let root = null;
for (let i = 0; i < n; i++) {
    root = insertNode(root, arr[i]);
}

console.log(maximumElement(root, a, b));
// THIS CODE IS CONTRIBUTED BY PIYUSH AGARWAL

Output:

12

Time complexity: O(n), where n is the number of nodes in the binary search tree.
Auxiliary Space: O(n)