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. We need to tell whether two BSTs will be identical or not without actually constructing the tree.

Example:

For example, the input arrays are {2, 4, 3, 1} and {2, 1, 4, 3} will construct the same tree 

 Let the input arrays be a[] and b[]

Example 1:
a[] = {2, 4, 1, 3} will construct following tree.
   2
 /  \
1    4
    /
   3
b[] = {2, 4, 3, 1} will also construct the same tree.
   2
 /  \
1    4
    /
   3 
So the output is "True"

Example 2:
a[] = {8, 3, 6, 1, 4, 7, 10, 14, 13}
b[] = {8, 10, 14, 3, 6, 4, 1, 7, 13}

They both construct the same following BST, so output is "True"
            8
         /    \
       3       10
     /  \        \
    1     6       14
        /   \     /
       4     7   13  

Solution:

• 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. The idea looks simple, but implementation requires checking all conditions for all elements. Following is an interesting recursive implementation of the idea. 

Implementation:

C++

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

/* The main function that checks if two
arrays a[] and b[] of size n construct
same BST. The two values 'min' and 'max'
decide whether the call is made for left
subtree or right subtree of a parent
element. The indexes i1 and i2 are the
indexes in (a[] and b[]) after which we
search the left or right child. Initially,
the call is made for INT_MIN and INT_MAX
as 'min' and 'max' respectively, because
root has no parent. i1 and i2 are just
after the indexes of the parent element in a[] and b[]. */
bool isSameBSTUtil(int a[], int b[], int n, int i1, int i2,
                   int min, int max)
{
    int j, k;

    /* Search for a value satisfying the
    constraints of min and max in a[] and
    b[]. If the parent element is a leaf
    node then there must be some elements
    in a[] and b[] satisfying constraint. */
    for (j = i1; j < n; j++)
        if (a[j] > min && a[j] < max)
            break;
    for (k = i2; k < n; k++)
        if (b[k] > min && b[k] < max)
            break;

    /* If the parent element is leaf in both arrays */
    if (j == n && k == n)
        return true;

    /* Return false if any of the following is true
        a) If the parent element is leaf in one array,
            but non-leaf in other.
        b) The elements satisfying constraints are
            not same. We either search for left
            child or right child of the parent
            element (decided by min and max values).
            The child found must be same in both arrays */
    if (((j == n) ^ (k == n)) || a[j] != b[k])
        return false;

    /* Make the current child as parent and
    recursively check for left and right
    subtrees of it. Note that we can also
    pass a[k] in place of a[j] as they
    are both are same */
    return isSameBSTUtil(a, b, n, j + 1, k + 1, a[j], max)
           && // Right Subtree
           isSameBSTUtil(a, b, n, j + 1, k + 1, min,
                         a[j]); // Left Subtree
}

// A wrapper over isSameBSTUtil()
bool isSameBST(int a[], int b[], int n)
{
    return isSameBSTUtil(a, b, n, 0, 0, INT_MIN, INT_MAX);
}

// Driver code
int main()
{
    int a[] = { 8, 3, 6, 1, 4, 7, 10, 14, 13 };
    int b[] = { 8, 10, 14, 3, 6, 4, 1, 7, 13 };
    int n = sizeof(a) / sizeof(a[0]);
    if (isSameBST(a, b, n)) {
        cout << "BSTs are same";
    }
    else {
        cout << "BSTs not same";
    }
    return 0;
}

// This code is contributed by rathbhupendra

C

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

/* The main function that checks if two arrays a[] and b[] of size n construct
  same BST. The two values 'min' and 'max' decide whether the call is made for
  left subtree or right subtree of a parent element. The indexes i1 and i2 are
  the indexes in (a[] and b[]) after which we search the left or right child.
  Initially, the call is made for INT_MIN and INT_MAX as 'min' and 'max'
  respectively, because root has no parent.
  i1 and i2 are just after the indexes of the parent element in a[] and b[]. */
bool isSameBSTUtil(int a[], int b[], int n, int i1, int i2, int min, int max)
{
   int j, k;

   /* Search for a value satisfying the constraints of min and max in a[] and
      b[]. If the parent element is a leaf node then there must be some
      elements in a[] and b[] satisfying constraint. */
   for (j=i1; j<n; j++)
       if (a[j]>min && a[j]<max)
           break;
   for (k=i2; k<n; k++)
       if (b[k]>min && b[k]<max)
           break;

   /* If the parent element is leaf in both arrays */
   if (j==n && k==n)
       return true;

   /* Return false if any of the following is true
      a) If the parent element is leaf in one array, but non-leaf in other.
      b) The elements satisfying constraints are not same. We either search
         for left child or right child of the parent element (decided by min
         and max values). The child found must be same in both arrays */
   if (((j==n)^(k==n)) || a[j]!=b[k])
       return false;

   /* Make the current child as parent and recursively check for left and right
      subtrees of it. Note that we can also pass a[k] in place of a[j] as they
      are both are same */
   return isSameBSTUtil(a, b, n, j+1, k+1, a[j], max) &&  // Right Subtree
          isSameBSTUtil(a, b, n, j+1, k+1, min, a[j]);    // Left Subtree
}

// A wrapper over isSameBSTUtil()
bool isSameBST(int a[], int b[], int n)
{
   return isSameBSTUtil(a, b, n, 0, 0, INT_MIN, INT_MAX);
}

// Driver program to test above functions
int main()
{
   int a[] = {8, 3, 6, 1, 4, 7, 10, 14, 13};
   int b[] = {8, 10, 14, 3, 6, 4, 1, 7, 13};
   int n=sizeof(a)/sizeof(a[0]);
   printf("%s\n", isSameBST(a, b, n)?
             "BSTs are same":"BSTs not same");
   return 0;
}

Java

// A Java program to check for Identical
// BSTs without building the trees
class GFG
{

/* The main function that checks
if two arrays a[] and b[] of size
n construct same BST. The two values
'min' and 'max' decide whether the
call is made for left subtree or
right subtree of a parent element.
The indexes i1 and i2 are the indexes
in (a[] and b[]) after which we search
the left or right child. Initially, the
call is made for INT_MIN and INT_MAX as
'min' and 'max' respectively, because
root has no parent. i1 and i2 are just
after the indexes of the parent element in a[] and b[]. */
static boolean isSameBSTUtil(int a[], int b[], int n,
                        int i1, int i2, int min, int max)
{
    int j, k;

    /* Search for a value satisfying the
    constraints of min and max in a[] and
    b[]. If the parent element is a leaf
    node then there must be some elements
    in a[] and b[] satisfying constraint. */
    for (j = i1; j < n; j++)
    if (a[j] > min && a[j] < max)
        break;
    for (k = i2; k < n; k++)
        if (b[k] > min && b[k] < max)
            break;

    /* If the parent element is
    leaf in both arrays */
    if (j == n && k == n)
        return true;

    /* Return false if any of the following is true
    a) If the parent element is leaf in
    one array, but non-leaf in other.
    b) The elements satisfying constraints
    are not same. We either search for left
    child or right child of the parent element
    (decided by min and max values). The child
    found must be same in both arrays */
    if (((j==n)^(k==n)) || a[j]!=b[k])
        return false;

    /* Make the current child as parent and
    recursively check for left and right
    subtrees of it. Note that we can also
    pass a[k] in place of a[j] as they
    are both are same */
    return isSameBSTUtil(a, b, n, j+1, k+1, a[j], max) && // Right Subtree
            isSameBSTUtil(a, b, n, j+1, k+1, min, a[j]); // Left Subtree
}

// A wrapper over isSameBSTUtil()
static boolean isSameBST(int a[], int b[], int n)
{
    return isSameBSTUtil(a, b, n, 0, 0,
                    Integer.MIN_VALUE,Integer.MAX_VALUE);
}

// Driver code
public static void main(String[] args)
{
    int a[] = {8, 3, 6, 1, 4, 7, 10, 14, 13};
    int b[] = {8, 10, 14, 3, 6, 4, 1, 7, 13};
    int n=a.length;
    System.out.printf("%s\n", isSameBST(a, b, n)?
                "BSTs are same":"BSTs not same");
}
}

/* This code contributed by PrinciRaj1992 */

Python3

# A Python3 program to check for Identical
# BSTs without building the trees

# # The main function that checks if two
# arrays a[] and b[] of size n construct
# same BST. The two values 'min' and 'max'
# decide whether the call is made for left
# subtree or right subtree of a parent
# element. The indexes i1 and i2 are the
# indexes in (a[] and b[]) after which we
# search the left or right child. Initially,
# the call is made for INT_MIN and INT_MAX
# as 'min' and 'max' respectively, because
# root has no parent. i1 and i2 are just
# after the indexes of the parent element in a[] and b[]. */
def isSameBSTUtil(a, b, n, i1, i2, min, max):

    # # Search for a value satisfying the
    # constraints of min and max in a[] and
    # b[]. If the parent element is a leaf
    # node then there must be some elements
    # in a[] and b[] satisfying constraint. */
    j, k = i1, i2
    while j < n:
        if (a[j] > min and a[j] < max):
            break;
        j += 1
    while k<n:
        if (b[k] > min and b[k] < max):
            break
        k += 1

    # If the parent element is leaf in both arrays */
    if (j == n and k == n):
        return True

    # Return false if any of the following is true
        # a) If the parent element is leaf in one array,
        #     but non-leaf in other.
        # b) The elements satisfying constraints are
        #     not same. We either search for left
        #     child or right child of the parent
        #     element (decided by min and max values).
        #     The child found must be same in both arrays */
    if (((j == n) ^ (k == n)) or a[j] != b[k]):
        return False

    # Make the current child as parent and
    # recursively check for left and right
    # subtrees of it. Note that we can also
    # pass a[k] in place of a[j] as they
    # are both are same */
    return isSameBSTUtil(a, b, n, j + 1, k + 1, a[j], max) and isSameBSTUtil(a, b, n, j + 1, k + 1, min, a[j]) #Left Subtree

# A wrapper over isSameBSTUtil()
def isSameBST(a, b, n):
    return isSameBSTUtil(a, b, n, 0, 0, -10**9, 10**9)

# Driver code
if __name__ == '__main__':
    a = [8, 3, 6, 1, 4, 7, 10, 14, 13]
    b = [8, 10, 14, 3, 6, 4, 1, 7, 13]
    n = len(a)

    if(isSameBST(a, b, n)):
        print("BSTs are same")
    else:
        print("BSTs not same")

# This code is contributed by mohit kumar 29.

C#

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

class GFG
{

/* The main function that checks
if two arrays a[] and b[] of size
n construct same BST. The two values
'min' and 'max' decide whether the
call is made for left subtree or
right subtree of a parent element.
The indexes i1 and i2 are the indexes
in (a[] and b[]) after which we search
the left or right child. Initially, the
call is made for INT_MIN and INT_MAX as
'min' and 'max' respectively, because
root has no parent. i1 and i2 are just
after the indexes of the parent element in a[] and b[]. */
static bool isSameBSTUtil(int []a, int []b, int n,
                        int i1, int i2, int min, int max)
{
    int j, k;

    /* Search for a value satisfying the
    constraints of min and max in a[] and
    b[]. If the parent element is a leaf
    node then there must be some elements
    in a[] and b[] satisfying constraint. */
    for (j = i1; j < n; j++)
    if (a[j] > min && a[j] < max)
        break;
    for (k = i2; k < n; k++)
        if (b[k] > min && b[k] < max)
            break;

    /* If the parent element is
    leaf in both arrays */
    if (j == n && k == n)
        return true;

    /* Return false if any of the following is true
    a) If the parent element is leaf in
    one array, but non-leaf in other.
    b) The elements satisfying constraints
    are not same. We either search for left
    child or right child of the parent element
    (decided by min and max values). The child
    found must be same in both arrays */
    if (((j == n)^(k == n)) || a[j] != b[k])
        return false;

    /* Make the current child as parent and
    recursively check for left and right
    subtrees of it. Note that we can also
    pass a[k] in place of a[j] as they
    are both are same */
    return isSameBSTUtil(a, b, n, j + 1, k + 1, a[j], max) && // Right Subtree
            isSameBSTUtil(a, b, n, j + 1, k + 1, min, a[j]); // Left Subtree
}

// A wrapper over isSameBSTUtil()
static bool isSameBST(int []a, int []b, int n)
{
    return isSameBSTUtil(a, b, n, 0, 0,
                    int.MinValue,int.MaxValue);
}

// Driver code
public static void Main(String[] args)
{
    int []a = {8, 3, 6, 1, 4, 7, 10, 14, 13};
    int []b = {8, 10, 14, 3, 6, 4, 1, 7, 13};
    int n=a.Length;
    Console.WriteLine("{0}\n", isSameBST(a, b, n)?
                "BSTs are same":"BSTs not same");
}
}

// This code has been contributed by 29AjayKumar

Javascript

<script>

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

    /* The main function that checks
if two arrays a[] and b[] of size
n construct same BST. The two values
'min' and 'max' decide whether the
call is made for left subtree or
right subtree of a parent element.
The indexes i1 and i2 are the indexes
in (a[] and b[]) after which we search
the left or right child. Initially, the
call is made for INT_MIN and INT_MAX as
'min' and 'max' respectively, because
root has no parent. i1 and i2 are just
after the indexes of the parent element in a[] and b[]. */
    function isSameBSTUtil(a,b,n,i1,i2,min,max)
    {
        let j, k;

    /* Search for a value satisfying the
    constraints of min and max in a[] and
    b[]. If the parent element is a leaf
    node then there must be some elements
    in a[] and b[] satisfying constraint. */
    for (j = i1; j < n; j++)
    if (a[j] > min && a[j] < max)
        break;
    for (k = i2; k < n; k++)
        if (b[k] > min && b[k] < max)
            break;

    /* If the parent element is
    leaf in both arrays */
    if (j == n && k == n)
        return true;

    /* Return false if any of the following is true
    a) If the parent element is leaf in
    one array, but non-leaf in other.
    b) The elements satisfying constraints
    are not same. We either search for left
    child or right child of the parent element
    (decided by min and max values). The child
    found must be same in both arrays */
    if (((j==n)^(k==n)) || a[j]!=b[k])
        return false;

    /* Make the current child as parent and
    recursively check for left and right
    subtrees of it. Note that we can also
    pass a[k] in place of a[j] as they
    are both are same */
    return isSameBSTUtil(a, b, n, j+1, k+1, a[j], max) && // Right Subtree
            isSameBSTUtil(a, b, n, j+1, k+1, min, a[j]); // Left Subtree
    }

    // A wrapper over isSameBSTUtil()
    function isSameBST(a,b,n)
    {
        return isSameBSTUtil(a, b, n, 0, 0,
                    Number.MIN_VALUE,Number.MAX_VALUE);
    }

    // Driver code
    let a=[8, 3, 6, 1, 4, 7, 10, 14, 13];
    let b=[8, 10, 14, 3, 6, 4, 1, 7, 13];
    let n=a.length;
    document.write( isSameBST(a, b, n)?
                "BSTs are same":"BSTs not same");

    // This code is contributed by unknown2108
</script>

Output

BSTs are same

Time Complexity: O(N²)
Auxiliary Space: O(N), for recursive stack space.