Check for Identical BSTs without building the trees

Given two arrays that represent a sequence of keys. Imagine we make a Binary Search Tree (BST) from each array. The task is to find whether two BSTs will be identical or not without actually constructing the tree.

Examples:

Input: arr1[] = {2, 4, 1, 3} , arr2[] = {2, 4, 3, 1}
Output: True
Explanation: As we can see the input arrays are {2, 4, 3, 1} and {2, 1, 4, 3} will construct the same tree

Input: arr1[] = {8, 3, 6, 1, 4, 7, 10, 14, 13}, arr2[] = {8, 10, 14, 3, 6, 4, 1, 7, 13}
Output: True
Explanation: They both construct the same following BST.

Approach:

According to BST property, elements of the left subtree must be smaller and elements of right subtree must be greater than root. Two arrays represent the same BST if, for every element x, the elements in left and right subtrees of x appear after it in both arrays. And same is true for roots of left and right subtrees. The idea is to check of , if next smaller and greater elements are same in both arrays. Same properties are recursively checked for left and right subtrees.

Follow the steps below to solve the problem:

In both arrays by comparing values within a range defined by min and max constraints. This starts from the root node.
For both arrays, find the next element that satisfies the current range of values (greater than min and less than max) to identify valid children.
If no valid children are found in both arrays, return true as both subtrees are leaves.
If one array has a valid child and the other does not, or if the children found are not equal, return false.
Move to the next valid child and recursively check both left and right subtrees by updating min and max values based on the current child.
Combine the results of the recursive calls for both left and right subtrees, returning true only if both match at every step.

Below is the implementation of the above approach:

C++

// C++ program to check for Identical
// BSTs without building the trees
#include <bits/stdc++.h>
using namespace std;

// Function to check if two arrays construct the same BST
bool isSameBSTUtil(vector<int>& arr1, vector<int>& arr2,  
                 int n, int l, int r, int low, int high) {
    int leftIdx, rightIdx;

    // Search for a valid element in arr1 and arr2
    // that satisfies the constraints `low` and `high`
    for (leftIdx = l; leftIdx < n; leftIdx++) {
        if (arr1[leftIdx] > low && arr1[leftIdx] < high) {
            break;
        }
    }
    for (rightIdx = r; rightIdx < n; rightIdx++) {
        if (arr2[rightIdx] > low && arr2[rightIdx] < high) {
            break;
        }
    }

    // If both arrays have no valid elements, return true
    if (leftIdx == n && rightIdx == n) {
        return true;
    }

    // If one array has a valid element but the other doesn't,
    // or if the elements are different, return false
    if (((leftIdx == n) ^ (rightIdx == n)) 
                     || arr1[leftIdx] != arr2[rightIdx]) {
      
        return false;
    }

    // Recursively check for the right and left subtrees
    return isSameBSTUtil(arr1, arr2, n, leftIdx + 1, rightIdx + 1, 
                         arr1[leftIdx], high) &&  
           isSameBSTUtil(arr1, arr2, n, leftIdx + 1, rightIdx + 1, 
                         low, arr1[leftIdx]);     
}

bool isSameBST(vector<int>& arr1, vector<int>& arr2,
                                               int n) {
    return isSameBSTUtil(arr1, arr2, n, 0, 0, 
                                INT_MIN, INT_MAX);
}

int main() {
  
    // Hardcoded inputs for both the arrays
    vector<int> arr1 = {8, 3, 6, 1, 4, 7, 10, 14, 13};
    vector<int> arr2 = {8, 10, 14, 3, 6, 4, 1, 7, 13};
    int n = arr1.size();

    if (isSameBST(arr1, arr2, n)) {
        cout << "True";
    } 
    else {
        cout << "False";
    }

    return 0;
}

// C program to check for Identical
// BSTs without building the trees
#include <stdio.h>
#include <limits.h>
#include <stdbool.h>

// Function to check if two arrays construct the same BST
bool isSameBSTUtil(int arr1[], int arr2[], int n, int l, 
                   int r, int low, int high) {
    int leftIdx, rightIdx;

    // Find the next element in bounds in arr1[]
    for (leftIdx = l; leftIdx < n; leftIdx++) {
        if (arr1[leftIdx] > low && arr1[leftIdx] < high)
            break;
    }

    // Find the next element in bounds in arr2[]
    for (rightIdx = r; rightIdx < n; rightIdx++) {
        if (arr2[rightIdx] > low && arr2[rightIdx] < high)
            break;
    }

    // If no valid elements in both arrays
    if (leftIdx == n && rightIdx == n)
        return true;

    // Mismatch in structure or values
    if (((leftIdx == n) ^ (rightIdx == n)) ||
        arr1[leftIdx] != arr2[rightIdx])
        return false;

    // Recur for left and right subtrees
    return isSameBSTUtil(arr1, arr2, n, leftIdx + 1, 
                         rightIdx + 1, arr1[leftIdx], 
                         high) &&  
           isSameBSTUtil(arr1, arr2, n, leftIdx + 1, 
                         rightIdx + 1, low, arr1[leftIdx]); 
}

bool isSameBST(int arr1[], int arr2[], int n) {
    return isSameBSTUtil(arr1, arr2, n, 0, 0, INT_MIN, 
                         INT_MAX);
}

int main() {
  
    // Hardcoded inputs for both the arrays
    int arr1[] = {8, 3, 6, 1, 4, 7, 10, 14, 13};
    int arr2[] = {8, 10, 14, 3, 6, 4, 1, 7, 13};
    int n = sizeof(arr1) / sizeof(arr1[0]);

    printf("%s\n", isSameBST(arr1, arr2, n) 
                  ? "True" 
                  : "False");

    return 0;
}

Java

// Java program to check for Identical
// BSTs without building the trees
import java.util.ArrayList;

class GfG {

    // Function to check if two arrays construct the same BST
    static boolean isSameBSTUtil(ArrayList<Integer> arr1,
        ArrayList<Integer> arr2, int n, int l,
                              int r, int low, int high) {

        int leftIdx, rightIdx;

        // Find the next element within bounds 
        // in arr1[] and arr2[]
        for (leftIdx = l; leftIdx < n; leftIdx++) {
            if (arr1.get(leftIdx) >
                 low && arr1.get(leftIdx) < high)
                break;
        }

        for (rightIdx = r; rightIdx < n; rightIdx++) {
            if (arr2.get(rightIdx) > 
                 low && arr2.get(rightIdx) < high)
                break;
        }

        // If no valid element is found in 
      	// both ArrayLists
        if (leftIdx == n && rightIdx == n)
            return true;

        // Mismatch in structure or values
        if (((leftIdx == n) ^ (rightIdx == n))
            || !arr1.get(leftIdx).equals(arr2.get(rightIdx)))
          
            return false;

        // Recur for left and right subtrees
        return isSameBSTUtil(arr1, arr2, n, leftIdx + 1, 
                 rightIdx + 1, arr1.get(leftIdx), high) && 
               isSameBSTUtil(arr1, arr2, n, leftIdx + 1, 
                    rightIdx + 1, low, arr1.get(leftIdx));  
    }

    // Wrapper over isSameBSTUtil()
    static boolean isSameBST(ArrayList<Integer> arr1,
                             ArrayList<Integer> arr2, int n) {
        return isSameBSTUtil(arr1, arr2, n, 0, 0,
            Integer.MIN_VALUE, Integer.MAX_VALUE);
    }

    public static void main(String[] args) {
      
        ArrayList<Integer> arr1 = new ArrayList<>();
        ArrayList<Integer> arr2 = new ArrayList<>();

        // Directly assigning elements to ArrayLists
        arr1.add(8); arr1.add(3); arr1.add(6);
        arr1.add(1); arr1.add(4); arr1.add(7);
        arr1.add(10); arr1.add(14); arr1.add(13);

        arr2.add(8); arr2.add(10); arr2.add(14);
        arr2.add(3); arr2.add(6); arr2.add(4);
        arr2.add(1); arr2.add(7); arr2.add(13);

        int n = arr1.size();

        System.out.println(isSameBST(arr1, arr2, n) ? 
            "True" : "False");
    }
}

Python

# A Python program to check for Identical
# BSTs without building the trees

# Function to check if two arrays construct the same BST
def isSameBSTUtil(arr1, arr2, n, l, r, low, high):
    
    # Find the next element within bounds in arr1[] and arr2[]
    while l < n:
        if low < arr1[l] < high:
            break
        l += 1

    while r < n:
        if low < arr2[r] < high:
            break
        r += 1

    # If no valid element is found in both arrays
    if l == n and r == n:
        return True

    # Mismatch in structure or values
    if (l == n) ^ (r == n) or arr1[l] != arr2[r]:
        return False

    # Recur for left and right subtrees
    return (isSameBSTUtil(arr1, arr2, n, l + 1, r + 1, arr1[l], high) and
            isSameBSTUtil(arr1, arr2, n, l + 1, r + 1, low, arr1[l]))


# A wrapper over isSameBSTUtil()
def isSameBST(arr1, arr2, n):
    return isSameBSTUtil(arr1, arr2, n, 0, 0, float('-inf'), float('inf'))


if __name__ == '__main__':
  
    # Hardcoded inputs for both the arrays
    arr1 = [8, 3, 6, 1, 4, 7, 10, 14, 13]
    arr2 = [8, 10, 14, 3, 6, 4, 1, 7, 13]
    n = len(arr1)

    print("True" if isSameBST(arr1, arr2, n) else "False")

// C# program to check for Identical 
// BSTs without building the trees 
using System;
using System.Collections.Generic;

class GfG { 
  
    // Function to check if two arrays construct the same BST
    static bool isSameBSTUtil(List<int> arr1, List<int> arr2,
                     int n, int l, int r, int low, int high) { 
        int leftIdx, rightIdx; 

        // Search for a valid element in arr1 and arr2
        // that satisfies the constraints `low` and `high`
        for (leftIdx = l; leftIdx < n; leftIdx++) 
            if (arr1[leftIdx] > low && arr1[leftIdx] < high) 
                break; 

        for (rightIdx = r; rightIdx < n; rightIdx++) 
            if (arr2[rightIdx] > low && arr2[rightIdx] < high) 
                break; 

        // If both arrays have no valid elements, return true
        if (leftIdx == n && rightIdx == n) 
            return true; 

        // If one array has a valid element but 
        // the other doesn't, or if the
        // elements are different, return false
        if (((leftIdx == n) ^ (rightIdx == n)) ||
                       arr1[leftIdx] != arr2[rightIdx]) 
            return false; 

        // Make the current child as parent and 
        // recursively check for left and right 
        // subtrees of it.
        return isSameBSTUtil(arr1, arr2, n, leftIdx + 1, 
                    rightIdx + 1, arr1[leftIdx], high) && 
               isSameBSTUtil(arr1, arr2, n, leftIdx + 1, 
                        rightIdx + 1, low, arr1[leftIdx]);
    } 

    // A wrapper over isSameBSTUtil() 
    static bool isSameBST(List<int> arr1,
                          List<int> arr2, int n) { 
      
        return isSameBSTUtil(arr1, arr2, n, 0, 0, 
                             int.MinValue, int.MaxValue); 
    } 

    static void Main(string[] args) { 
      
        // Hardcoded Inputs for the Arrays
        List<int> arr1 
          = new List<int> {8, 3, 6, 1, 4, 7, 10, 14, 13}; 
        List<int> arr2
          = new List<int> {8, 10, 14, 3, 6, 4, 1, 7, 13}; 
        int n = arr1.Count; 

        Console.WriteLine(isSameBST(arr1, arr2, n) ? 
                          "True" : "False"); 
    } 
}

JavaScript

// Javascript program to check for Identical
// BSTs without building the trees

// Function to check if two arrays construct the same BST
function isSameBSTUtil(arr1, arr2, n, l, r, low, high) {
    let leftIdx, rightIdx;

    // Find valid elements in arr1 and arr2 within bounds
    for (leftIdx = l; leftIdx < n; leftIdx++) {
        if (arr1[leftIdx] > low && 
                       arr1[leftIdx] < high) break;
    }
    for (rightIdx = r; rightIdx < n; rightIdx++) {
        if (arr2[rightIdx] > low && 
                       arr2[rightIdx] < high) break;
    }

    // Both arrays have reached the end
    if (leftIdx == n && rightIdx == n) return true;

    // Mismatch in structure or values
    if (((leftIdx == n) ^ (rightIdx == n)) ||
              arr1[leftIdx] != arr2[rightIdx]) 
       return false;

    // Recursively check left and right subtrees
    return isSameBSTUtil(arr1, arr2, n, leftIdx + 1,
                  rightIdx + 1, arr1[leftIdx], high) &&
           isSameBSTUtil(arr1, arr2, n, leftIdx + 1, 
           rightIdx + 1, low, arr1[leftIdx]);
}

// Wrapper function
function isSameBST(arr1, arr2, n) {
    return isSameBSTUtil(arr1, arr2, n, 0, 0, 
                   Number.MIN_VALUE, Number.MAX_VALUE);
}

// Harcoded inputs for both arrays
let arr1 = [8, 3, 6, 1, 4, 7, 10, 14, 13];
let arr2 = [8, 10, 14, 3, 6, 4, 1, 7, 13];
let n = arr1.length;
console.log(isSameBST(arr1, arr2, n) ? 
                   "True" : "False");

Output

True

Time Complexity: O(n^2), because for each node, we recursively search through the rest of the arrays to find the corresponding nodes in both arrays.
Auxiliary Space: O(h), where h is the height of tree.

Check for Identical BSTs without building the trees

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