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Check if given sorted sub-sequence exists in binary search tree

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Given a binary search tree and a sorted sub-sequence. the task is to check if the given sorted sub-sequence exist in binary search tree or not.
 

200px-Binary_search_tree.svg


Examples: 
 

// For above binary search tree
Input : seq[] = {4, 6, 8, 14}
Output: "Yes"

Input : seq[] = {4, 6, 8, 12, 13}
Output: "No"


 

A simple solution is to store inorder traversal in an auxiliary array and then by matching elements of sorted sub-sequence one by one with inorder traversal of tree , we can if sub-sequence exist in BST or not. Time complexity for this approach will be O(n) but it requires extra space O(n) for storing traversal in an array.

An efficient solution is to match elements of sub-sequence while we are traversing BST in inorder fashion. We take index as a iterator for given sorted sub-sequence and start inorder traversal of given bst, if current node matches with seq[index] then move index in forward direction by incrementing 1 and after complete traversal of BST if index==n that means all elements of given sub-sequence have been matched and exist as a sorted sub-sequence in given BST. 
 

C++

// C++ program to find if given array exists as a
// subsequence in BST
#include<bits/stdc++.h>
using namespace std;
  
// A binary Tree node
struct Node
{
    int data;
    struct Node *left, *right;
};
  
// A utility function to create a new BST node
// with key as given num
struct Node* newNode(int num)
{
    struct Node* temp = new Node;
    temp->data = num;
    temp->left = temp->right = NULL;
    return temp;
}
  
// A utility function to insert a given key to BST
struct Node* insert(struct Node* root, int key)
{
    if (root == NULL)
        return newNode(key);
    if (root->data > key)
        root->left = insert(root->left, key);
    else
        root->right = insert(root->right, key);
    return root;
}
  
// function to check if given sorted sub-sequence exist in BST
// index --> iterator for given sorted sub-sequence
// seq[] --> given sorted sub-sequence
void seqExistUtil(struct Node *ptr, int seq[], int &index)
{
    if (ptr == NULL)
        return;
  
    // We traverse left subtree first in Inorder
    seqExistUtil(ptr->left, seq, index);
  
    // If current node matches with se[index] then move
    // forward in sub-sequence
    if (ptr->data == seq[index])
        index++;
  
    // We traverse left subtree in the end in Inorder
    seqExistUtil(ptr->right, seq, index);
}
  
// A wrapper over seqExistUtil. It returns true
// if seq[0..n-1] exists in tree.
bool seqExist(struct Node *root, int seq[], int n)
{
    // Initialize index in seq[]
    int index = 0;
  
    // Do an inorder traversal and find if all
    // elements of seq[] were present
    seqExistUtil(root, seq, index);
  
    // index would become n if all elements of
    // seq[] were present
    return (index == n);
}
  
// driver program to run the case
int main()
{
    struct Node* root = NULL;
    root = insert(root, 8);
    root = insert(root, 10);
    root = insert(root, 3);
    root = insert(root, 6);
    root = insert(root, 1);
    root = insert(root, 4);
    root = insert(root, 7);
    root = insert(root, 14);
    root = insert(root, 13);
  
    int seq[] = {4, 6, 8, 14};
    int n = sizeof(seq)/sizeof(seq[0]);
  
    seqExist(root, seq, n)? cout << "Yes" :
                            cout << "No";
  
    return 0;
}

                    

Java

// Java program to find if given array 
// exists as a subsequence in BST 
import java.util.*;
  
class GFG
{
      
// A binary Tree node 
static class Node 
    int data; 
    Node left, right; 
}; 
  
//structure of int class
static class INT
{
    int a;
}
  
// A utility function to create a new BST node 
// with key as given num 
static Node newNode(int num) 
    Node temp = new Node(); 
    temp.data = num; 
    temp.left = temp.right = null
    return temp; 
  
// A utility function to insert a given key to BST 
static Node insert( Node root, int key) 
    if (root == null
        return newNode(key); 
    if (root.data > key) 
        root.left = insert(root.left, key); 
    else
        root.right = insert(root.right, key); 
    return root; 
  
// function to check if given sorted 
// sub-sequence exist in BST index -. 
// iterator for given sorted sub-sequence 
// seq[] -. given sorted sub-sequence 
static void seqExistUtil( Node ptr, int seq[], INT index) 
    if (ptr == null
        return
  
    // We traverse left subtree 
    // first in Inorder 
    seqExistUtil(ptr.left, seq, index); 
  
    // If current node matches 
    // with se[index] then move 
    // forward in sub-sequence 
    if (ptr.data == seq[index.a]) 
        index.a++; 
  
    // We traverse left subtree
    // in the end in Inorder 
    seqExistUtil(ptr.right, seq, index); 
  
// A wrapper over seqExistUtil.
// It returns true if seq[0..n-1] 
// exists in tree. 
static boolean seqExist( Node root, int seq[], int n) 
    // Initialize index in seq[] 
    INT index = new INT();
      
    index.a = 0;
  
    // Do an inorder traversal and find if all 
    // elements of seq[] were present 
    seqExistUtil(root, seq, index); 
  
    // index would become n if all 
    // elements of seq[] were present 
    return (index.a == n); 
  
// Driver code 
public static void main(String args[]) 
    Node root = null
    root = insert(root, 8); 
    root = insert(root, 10); 
    root = insert(root, 3); 
    root = insert(root, 6); 
    root = insert(root, 1); 
    root = insert(root, 4); 
    root = insert(root, 7); 
    root = insert(root, 14); 
    root = insert(root, 13); 
  
    int seq[] = {4, 6, 8, 14}; 
    int n = seq.length; 
  
    if(seqExist(root, seq, n))
        System.out.println("Yes");
    else
        System.out.println("No");
}
  
// This code is contributed by Arnab Kundu

                    

Python3

# Python3 program to find if given array
# exists as a subsequence in BST
class Node: 
  
    # Constructor to create a new node 
    def __init__(self, data): 
        self.data = data 
        self.left = None
        self.right = None
          
# A utility function to insert a 
# given key to BST 
def insert(root, key):
    if root == None
        return Node(key) 
    if root.data > key: 
        root.left = insert(root.left, key) 
    else:
        root.right = insert(root.right, key) 
    return root
  
# function to check if given sorted 
# sub-sequence exist in BST index .
# iterator for given sorted sub-sequence 
# seq[] . given sorted sub-sequence 
def seqExistUtil(ptr, seq, index):
    if ptr == None
        return
  
    # We traverse left subtree 
    # first in Inorder 
    seqExistUtil(ptr.left, seq, index) 
  
    # If current node matches with se[index[0]] 
    # then move forward in sub-sequence 
    if ptr.data == seq[index[0]]: 
        index[0] += 1
  
    # We traverse left subtree in
    # the end in Inorder 
    seqExistUtil(ptr.right, seq, index)
  
# A wrapper over seqExistUtil. It returns 
# true if seq[0..n-1] exists in tree. 
def seqExist(root, seq, n):
      
    # Initialize index in seq[] 
    index = [0]
  
    # Do an inorder traversal and find if 
    # all elements of seq[] were present 
    seqExistUtil(root, seq, index)
  
    # index would become n if all elements
    # of seq[] were present 
    if index[0] == n:
        return True
    else:
        return False
  
# Driver Code
if __name__ == '__main__':
    root = None
    root = insert(root, 8)
    root = insert(root, 10)
    root = insert(root, 3)
    root = insert(root, 6)
    root = insert(root, 1)
    root = insert(root, 4)
    root = insert(root, 7)
    root = insert(root, 14)
    root = insert(root, 13)
  
    seq = [4, 6, 8, 14
    n = len(seq)
    if seqExist(root, seq, n):
        print("Yes")
    else:
        print("No"
  
# This code is contributed by PranchalK

                    

C#

// C# program to find if given array 
// exists as a subsequence in BST 
using System;
  
class GFG 
      
// A binary Tree node 
public class Node 
    public int data; 
    public Node left, right; 
}; 
  
// structure of int class 
public class INT 
    public int a; 
  
// A utility function to create a new BST node 
// with key as given num 
static Node newNode(int num) 
    Node temp = new Node(); 
    temp.data = num; 
    temp.left = temp.right = null
    return temp; 
  
// A utility function to insert a given key to BST 
static Node insert( Node root, int key) 
    if (root == null
        return newNode(key); 
    if (root.data > key) 
        root.left = insert(root.left, key); 
    else
        root.right = insert(root.right, key); 
    return root; 
  
// function to check if given sorted 
// sub-sequence exist in BST index -. 
// iterator for given sorted sub-sequence 
// seq[] -. given sorted sub-sequence 
static void seqExistUtil( Node ptr, int []seq, INT index) 
    if (ptr == null
        return
  
    // We traverse left subtree 
    // first in Inorder 
    seqExistUtil(ptr.left, seq, index); 
  
    // If current node matches 
    // with se[index] then move 
    // forward in sub-sequence 
    if (ptr.data == seq[index.a]) 
        index.a++; 
  
    // We traverse left subtree 
    // in the end in Inorder 
    seqExistUtil(ptr.right, seq, index); 
  
// A wrapper over seqExistUtil. 
// It returns true if seq[0..n-1] 
// exists in tree. 
static bool seqExist( Node root, int []seq, int n) 
    // Initialize index in seq[] 
    INT index = new INT(); 
      
    index.a = 0; 
  
    // Do an inorder traversal and find if all 
    // elements of seq[] were present 
    seqExistUtil(root, seq, index); 
  
    // index would become n if all 
    // elements of seq[] were present 
    return (index.a == n); 
  
// Driver code 
public static void Main(String []args) 
    Node root = null
    root = insert(root, 8); 
    root = insert(root, 10); 
    root = insert(root, 3); 
    root = insert(root, 6); 
    root = insert(root, 1); 
    root = insert(root, 4); 
    root = insert(root, 7); 
    root = insert(root, 14); 
    root = insert(root, 13); 
  
    int []seq = {4, 6, 8, 14}; 
    int n = seq.Length; 
  
    if(seqExist(root, seq, n)) 
        Console.WriteLine("Yes"); 
    else
        Console.WriteLine("No"); 
  
/* This code contributed by PrinciRaj1992 */

                    

Javascript

<script>
  
// JavaScript program to find if given array 
// exists as a subsequence in BST 
  
      
// A binary Tree node 
class Node 
    constructor()
    {
        this.data = 0;
        this.left = null;
        this.right = null;
    }
}; 
  
// structure of int class 
class INT 
    constructor()
    {
        this.a = 0;
    }
  
// A utility function to create a new BST node 
// with key as given num 
function newNode(num) 
    var temp = new Node(); 
    temp.data = num; 
    temp.left = temp.right = null
    return temp; 
  
// A utility function to insert a given key to BST 
function insert( root, key) 
    if (root == null
        return newNode(key); 
    if (root.data > key) 
        root.left = insert(root.left, key); 
    else
        root.right = insert(root.right, key); 
    return root; 
  
// function to check if given sorted 
// sub-sequence exist in BST index -. 
// iterator for given sorted sub-sequence 
// seq[] -. given sorted sub-sequence 
function seqExistUtil( ptr, seq, index) 
    if (ptr == null
        return
  
    // We traverse left subtree 
    // first in Inorder 
    seqExistUtil(ptr.left, seq, index); 
  
    // If current node matches 
    // with se[index] then move 
    // forward in sub-sequence 
    if (ptr.data == seq[index.a]) 
        index.a++; 
  
    // We traverse left subtree 
    // in the end in Inorder 
    seqExistUtil(ptr.right, seq, index); 
  
// A wrapper over seqExistUtil. 
// It returns true if seq[0..n-1] 
// exists in tree. 
function seqExist( root, seq, n) 
    // Initialize index in seq[] 
    var index = new INT(); 
      
    index.a = 0; 
  
    // Do an inorder traversal and find if all 
    // elements of seq[] were present 
    seqExistUtil(root, seq, index); 
  
    // index would become n if all 
    // elements of seq[] were present 
    return (index.a == n); 
  
// Driver code 
var root = null
root = insert(root, 8); 
root = insert(root, 10); 
root = insert(root, 3); 
root = insert(root, 6); 
root = insert(root, 1); 
root = insert(root, 4); 
root = insert(root, 7); 
root = insert(root, 14); 
root = insert(root, 13); 
var seq = [4, 6, 8, 14]; 
var n = seq.length; 
if(seqExist(root, seq, n)) 
    document.write("Yes"); 
else
    document.write("No"); 
  
</script>

                    

Output
Yes

Time complexity: O(n)
Auxiliary Space: O(n) for call stack since using recursion, where n is total no of nodes in BST




 

Approach 2 :- Optimal solution in O(N) time complexity and O(h) space complexity 

The existsSubsequence function takes a BST root and a vector seq representing the sorted sub-sequence to be searched. It first checks if the seq is empty, in which case it returns true. Otherwise, it calls the findSequence function with the root, seq, and index i set to 0. The findSequence function recursively searches for the elements in the seq starting from index i in the BST. If the current node has the same value as the element at index i, it increments i and recursively calls the findSequence function on the left and right subtrees with the updated value of i. If i reaches the end of seq, it means that the sub-sequence exists in the BST, and the function returns true. If the current node does not have the same value as the element at index i, the function returns false.

C++

#include <iostream>
#include <vector>
  
using namespace std;
  
struct TreeNode {
    int val;
    TreeNode* left;
    TreeNode* right;
    TreeNode(int x) : val(x), left(NULL), right(NULL) {}
};
  
bool findSequence(TreeNode* root, vector<int>& seq, int i) {
    if (root == NULL) {
        return false;
    }
    if (root->val == seq[i]) {
        i++;
        if (i == seq.size()) {
            return true;
        }
        return findSequence(root->left, seq, i) || findSequence(root->right, seq, i);
    }
    return false;
}
  
bool existsSubsequence(TreeNode* root, vector<int>& seq) {
    if (seq.empty()) {
        return true;
    }
    return findSequence(root, seq, 0);
}
  
int main() {
    // create a BST
    TreeNode* root = new TreeNode(5);
    root->left = new TreeNode(3);
    root->right = new TreeNode(7);
    root->left->left = new TreeNode(2);
    root->left->right = new TreeNode(4);
    root->right->left = new TreeNode(6);
    root->right->right = new TreeNode(8);
  
    // check if a given sorted sub-sequence exists in the BST
    vector<int> seq1 = {2, 3, 4};
     
    if (existsSubsequence(root, seq1)) {
        cout << "The sequence {2, 3, 4} exists in the BST" << endl;
    } else {
        cout << "The sequence {2, 3, 4} does not exist in the BST" << endl;
    }
      
  
    return 0;
}

                    

Java

import java.util.ArrayList;
  
class TreeNode {
    int val;
    TreeNode left;
    TreeNode right;
      
    public TreeNode(int x) {
        val = x;
        left = null;
        right = null;
    }
}
  
public class Main {
    public static boolean findSequence(TreeNode root, ArrayList<Integer> seq, int i) {
        if (root == null) {
            return false;
        }
        if (root.val == seq.get(i)) {
            i++;
            if (i == seq.size()) {
                return true;
            }
            return findSequence(root.left, seq, i) || findSequence(root.right, seq, i);
        }
        return false;
    }
      
    public static boolean existsSubsequence(TreeNode root, ArrayList<Integer> seq) {
        if (seq.isEmpty()) {
            return true;
        }
        return findSequence(root, seq, 0);
    }
      
    public static void main(String[] args) {
        // create a BST
        TreeNode root = new TreeNode(5);
        root.left = new TreeNode(3);
        root.right = new TreeNode(7);
        root.left.left = new TreeNode(2);
        root.left.right = new TreeNode(4);
        root.right.left = new TreeNode(6);
        root.right.right = new TreeNode(8);
      
        // check if a given sorted sub-sequence exists in the BST
        ArrayList<Integer> seq1 = new ArrayList<>();
        seq1.add(2);
        seq1.add(3);
        seq1.add(4);
          
        if (existsSubsequence(root, seq1)) {
            System.out.println("The sequence {2, 3, 4} exists in the BST");
        } else {
            System.out.println("The sequence {2, 3, 4} does not exist in the BST");
        }
    }
}

                    

Python3

# javascript code implementation 
class Node:
      
    def __init__(self, val):
        self.val = val
        self.left = None
        self.right = None
      
  
def findSequence(root, seq, i):
    if (root == None):
        return False
      
    if (root.val == seq[i]):
        i = i + 1
        if (i == len(seq)):
            return True
        return findSequence(root.left, seq, i) or findSequence(root.right, seq, i)
    return False
  
  
def existsSubsequence(root, seq):
    if (len(seq) == 0):
        return True
    return findSequence(root, seq, 0)
  
  
root = Node(5)
root.left = Node(3)
root.right = Node(7)
root.left.left = Node(2)
root.left.right = Node(4)
root.right.left = Node(6)
root.right.right = Node(8)
  
# check if a given sorted sub-sequence exists in the BST
seq1 = [2, 3, 4]
  
if (existsSubsequence(root, seq1)):
    print("The sequence {2, 3, 4} exists in the BST")
else:
    print("The sequence {2, 3, 4} does not exist in the BST")
  
# The code is contributed by Nidhi goel. 

                    

C#

using System;
using System.Collections.Generic;
  
public class TreeNode {
    public int val;
    public TreeNode left;
    public TreeNode right;
    public TreeNode(int x) { val = x; }
}
  
public class Program {
    public static bool FindSequence(TreeNode root,
                                    List<int> seq, int i)
    {
        if (root == null) {
            return false;
        }
        if (root.val == seq[i]) {
            i++;
            if (i == seq.Count) {
                return true;
            }
            return FindSequence(root.left, seq, i)
                || FindSequence(root.right, seq, i);
        }
        return false;
    }
  
    public static bool ExistsSubsequence(TreeNode root,
                                         List<int> seq)
    {
        if (seq.Count == 0) {
            return true;
        }
        return FindSequence(root, seq, 0);
    }
  
    public static void Main()
    {
        // create a BST
        TreeNode root = new TreeNode(5);
        root.left = new TreeNode(3);
        root.right = new TreeNode(7);
        root.left.left = new TreeNode(2);
        root.left.right = new TreeNode(4);
        root.right.left = new TreeNode(6);
        root.right.right = new TreeNode(8);
  
        // check if a given sorted sub-sequence exists in
        // the BST
        List<int> seq1 = new List<int>{ 2, 3, 4 };
  
        if (ExistsSubsequence(root, seq1)) {
            Console.WriteLine(
                "The sequence {2, 3, 4} exists in the BST");
        }
        else {
            Console.WriteLine(
                "The sequence {2, 3, 4} does not exist in the BST");
        }
    }
}
// This code is contributed by user_dtewbxkn77n

                    

Javascript

// javascript code implementation 
class Node {
      
    constructor(val){
        this.val = val;
        this.left = null;
        this.right = null;
    }
}
  
function findSequence(root, seq, i) {
    if (root == null) {
        return false;
    }
    if (root.val == seq[i]) {
        i++;
        if (i == seq.length) {
            return true;
        }
        return findSequence(root.left, seq, i) || findSequence(root.right, seq, i);
    }
    return false;
}
  
function existsSubsequence(root, seq) {
    if (seq.length == 0) {
        return true;
    }
    return findSequence(root, seq, 0);
}
  
  
let root = new Node(5);
root.left = new Node(3);
root.right = new Node(7);
root.left.left = new Node(2);
root.left.right = new Node(4);
root.right.left = new Node(6);
root.right.right = new Node(8);
  
// check if a given sorted sub-sequence exists in the BST
let seq1 = [2, 3, 4];
  
if (existsSubsequence(root, seq1)) {
    console.log("The sequence {2, 3, 4} exists in the BST");
} else {
    console.log("The sequence {2, 3, 4} does not exist in the BST");
}
  
// The code is contributed by Arushi Jindal.

                    

Output
The sequence {2, 3, 4} does not exist in the BST

Time complexity –The time complexity of the given program depends on the height of the BST and the length of the input sequence. In the worst case, where the input sequence is a valid subsequence of the BST and has a length of k, and the height of the BST is h, the time complexity of the program will be O(k*h).

The reason for this is that the program performs a depth-first search on the BST to find the nodes in the input sequence, and the search may need to visit up to k nodes in the BST. Since the search is a recursive function that traverses the tree from the root to the leaves, the worst-case time complexity is O(k*h).

Space complexity – The space complexity of the program depends on the height of the BST and the length of the input sequence. In the worst case, where the input sequence is a valid subsequence of the BST and has a length of k, and the height of the BST is h, the space complexity of the program will be O(h).

The reason for this is that the program uses a recursive function to traverse the BST, and the maximum number of function calls that can be on the call stack at any given time is equal to the height of the BST. In the worst case, where the input sequence is a valid subsequence of the BST and has a length of k, and the height of the BST is h, the space complexity is O(h).



Last Updated : 11 Sep, 2023
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