# Merge two BSTs with limited extra space

Given two Binary Search Trees(BST), print the elements of both BSTs in sorted form. The expected time complexity is O(m+n) where m is the number of nodes in first tree and n is the number of nodes in second tree. Maximum allowed auxiliary space is O(height of the first tree + height of the second tree).

Examples:

```First BST
3
/
1       5
Second BST
4
/
2       6
Output: 1 2 3 4 5 6

First BST
8
/
2   10
/
1
Second BST
5
/
3
/
0
Output: 0 1 2 3 5 8 10
```

A similar question has been discussed earlier. Let us first discuss already discussed methods of the previous post which was for Balanced BSTs. The method 1 can be applied here also, but the time complexity will be O(n^2) in worst case. The method 2 can also be applied here, but the extra space required will be O(n) which violates the constraint given in this question. Method 3 can be applied here but the step 3 of method 3 can’t be done in O(n) for an unbalanced BST.

Thanks to Kumar for suggesting the following solution.

The idea is to use iterative inorder traversal. We use two auxiliary stacks for two BSTs. Since we need to print the elements in sorted form, whenever we get a smaller element from any of the trees, we print it. If the element is greater, then we push it back to stack for the next iteration.

## C++

 `#include ` `using` `namespace` `std; ` ` `  `// Structure of a BST Node  ` `class` `node  ` `{  ` `    ``public``: ` `    ``int` `data;  ` `    ``node *left;  ` `    ``node *right;  ` `};  ` ` `  `//.................... START OF STACK RELATED STUFF....................  ` `// A stack node  ` `class` `snode  ` `{  ` `    ``public``: ` `    ``node *t;  ` `    ``snode *next;  ` `};  ` ` `  `// Function to add an elemt k to stack  ` `void` `push(snode **s, node *k)  ` `{  ` `    ``snode *tmp = ``new` `snode();  ` ` `  `    ``//perform memory check here  ` `    ``tmp->t = k;  ` `    ``tmp->next = *s;  ` `    ``(*s) = tmp;  ` `}  ` ` `  `// Function to pop an element t from stack  ` `node *pop(snode **s)  ` `{  ` `    ``node *t;  ` `    ``snode *st;  ` `    ``st=*s;  ` `    ``(*s) = (*s)->next;  ` `    ``t = st->t;  ` `    ``free``(st);  ` `    ``return` `t;  ` `}  ` ` `  `// Fucntion to check whether the stack is empty or not  ` `int` `isEmpty(snode *s)  ` `{  ` `    ``if` `(s == NULL )  ` `        ``return` `1;  ` ` `  `    ``return` `0;  ` `}  ` `//.................... END OF STACK RELATED STUFF....................  ` ` `  ` `  `/* Utility function to create a new Binary Tree node */` `node* newNode (``int` `data)  ` `{  ` `    ``node *temp = ``new` `node;  ` `    ``temp->data = data;  ` `    ``temp->left = NULL;  ` `    ``temp->right = NULL;  ` `    ``return` `temp;  ` `}  ` ` `  `/* A utility function to print Inoder traversal of a Binary Tree */` `void` `inorder(node *root)  ` `{  ` `    ``if` `(root != NULL)  ` `    ``{  ` `        ``inorder(root->left);  ` `        ``cout<data<<``" "``;  ` `        ``inorder(root->right);  ` `    ``}  ` `}  ` ` `  `// The function to print data of two BSTs in sorted order  ` `void` `merge(node *root1, node *root2)  ` `{  ` `    ``// s1 is stack to hold nodes of first BST  ` `    ``snode *s1 = NULL;  ` ` `  `    ``// Current node of first BST  ` `    ``node *current1 = root1;  ` ` `  `    ``// s2 is stack to hold nodes of second BST  ` `    ``snode *s2 = NULL;  ` ` `  `    ``// Current node of second BST  ` `    ``node *current2 = root2;  ` ` `  `    ``// If first BST is empty, then output is inorder  ` `    ``// traversal of second BST  ` `    ``if` `(root1 == NULL)  ` `    ``{  ` `        ``inorder(root2);  ` `        ``return``;  ` `    ``}  ` `    ``// If second BST is empty, then output is inorder  ` `    ``// traversal of first BST  ` `    ``if` `(root2 == NULL)  ` `    ``{  ` `        ``inorder(root1);  ` `        ``return` `;  ` `    ``}  ` ` `  `    ``// Run the loop while there are nodes not yet printed.  ` `    ``// The nodes may be in stack(explored, but not printed)  ` `    ``// or may be not yet explored  ` `    ``while` `(current1 != NULL || !isEmpty(s1) ||  ` `        ``current2 != NULL || !isEmpty(s2))  ` `    ``{  ` `        ``// Following steps follow iterative Inorder Traversal  ` `        ``if` `(current1 != NULL || current2 != NULL )  ` `        ``{  ` `            ``// Reach the leftmost node of both BSTs and push ancestors of  ` `            ``// leftmost nodes to stack s1 and s2 respectively  ` `            ``if` `(current1 != NULL)  ` `            ``{  ` `                ``push(&s1, current1);  ` `                ``current1 = current1->left;  ` `            ``}  ` `            ``if` `(current2 != NULL)  ` `            ``{  ` `                ``push(&s2, current2);  ` `                ``current2 = current2->left;  ` `            ``}  ` ` `  `        ``}  ` `        ``else` `        ``{  ` `            ``// If we reach a NULL node and either of the stacks is empty,  ` `            ``// then one tree is exhausted, ptint the other tree  ` `            ``if` `(isEmpty(s1))  ` `            ``{  ` `                ``while` `(!isEmpty(s2))  ` `                ``{  ` `                    ``current2 = pop (&s2);  ` `                    ``current2->left = NULL;  ` `                    ``inorder(current2);  ` `                ``}  ` `                ``return` `;  ` `            ``}  ` `            ``if` `(isEmpty(s2))  ` `            ``{  ` `                ``while` `(!isEmpty(s1))  ` `                ``{  ` `                    ``current1 = pop (&s1);  ` `                    ``current1->left = NULL;  ` `                    ``inorder(current1);  ` `                ``}  ` `                ``return` `;  ` `            ``}  ` ` `  `            ``// Pop an element from both stacks and compare the  ` `            ``// popped elements  ` `            ``current1 = pop(&s1);  ` `            ``current2 = pop(&s2);  ` ` `  `            ``// If element of first tree is smaller, then print it  ` `            ``// and push the right subtree. If the element is larger,  ` `            ``// then we push it back to the corresponding stack.  ` `            ``if` `(current1->data < current2->data)  ` `            ``{  ` `                ``cout<data<<``" "``;  ` `                ``current1 = current1->right;  ` `                ``push(&s2, current2);  ` `                ``current2 = NULL;  ` `            ``}  ` `            ``else` `            ``{  ` `                ``cout<data<<``" "``;  ` `                ``current2 = current2->right;  ` `                ``push(&s1, current1);  ` `                ``current1 = NULL;  ` `            ``}  ` `        ``}  ` `    ``}  ` `}  ` ` `  `/* Driver program to test above functions */` `int` `main()  ` `{  ` `    ``node *root1 = NULL, *root2 = NULL;  ` ` `  `    ``/* Let us create the following tree as first tree  ` `            ``3  ` `        ``/   ` `        ``1 5  ` `    ``*/` `    ``root1 = newNode(3);  ` `    ``root1->left = newNode(1);  ` `    ``root1->right = newNode(5);  ` ` `  `    ``/* Let us create the following tree as second tree  ` `            ``4  ` `        ``/   ` `        ``2 6  ` `    ``*/` `    ``root2 = newNode(4);  ` `    ``root2->left = newNode(2);  ` `    ``root2->right = newNode(6);  ` ` `  `    ``// Print sorted nodes of both trees  ` `    ``merge(root1, root2);  ` ` `  `    ``return` `0;  ` `}  ` ` `  `//This code is contributed by rathbhupendra `

## C

 `#include ` `#include ` ` `  `// Structure of a BST Node ` `struct` `node ` `{ ` `    ``int` `data; ` `    ``struct` `node *left; ` `    ``struct` `node *right; ` `}; ` ` `  `//.................... START OF STACK RELATED STUFF.................... ` `// A stack node ` `struct` `snode ` `{ ` `    ``struct` `node  *t; ` `    ``struct` `snode *next; ` `}; ` ` `  `// Function to add an elemt k to stack ` `void` `push(``struct` `snode **s, ``struct` `node *k) ` `{ ` `    ``struct` `snode *tmp = (``struct` `snode *) ``malloc``(``sizeof``(``struct` `snode)); ` ` `  `    ``//perform memory check here ` `    ``tmp->t = k; ` `    ``tmp->next = *s; ` `    ``(*s) = tmp; ` `} ` ` `  `// Function to pop an element t from stack ` `struct` `node *pop(``struct` `snode **s) ` `{ ` `    ``struct`  `node *t; ` `    ``struct` `snode *st; ` `    ``st=*s; ` `    ``(*s) = (*s)->next; ` `    ``t = st->t; ` `    ``free``(st); ` `    ``return` `t; ` `} ` ` `  `// Fucntion to check whether the stack is empty or not ` `int` `isEmpty(``struct` `snode *s) ` `{ ` `    ``if` `(s == NULL ) ` `        ``return` `1; ` ` `  `    ``return` `0; ` `} ` `//.................... END OF STACK RELATED STUFF.................... ` ` `  ` `  `/* Utility function to create a new Binary Tree node */` `struct` `node* newNode (``int` `data) ` `{ ` `    ``struct` `node *temp = (``struct` `node*)``malloc``(``sizeof``(``struct` `node)); ` `    ``temp->data = data; ` `    ``temp->left = NULL; ` `    ``temp->right = NULL; ` `    ``return` `temp; ` `} ` ` `  `/* A utility function to print Inoder traversal of a Binary Tree */` `void` `inorder(``struct` `node *root) ` `{ ` `    ``if` `(root != NULL) ` `    ``{ ` `        ``inorder(root->left); ` `        ``printf``(``"%d "``, root->data); ` `        ``inorder(root->right); ` `    ``} ` `} ` ` `  `// The function to print data of two BSTs in sorted order ` `void`  `merge(``struct` `node *root1, ``struct` `node *root2) ` `{ ` `    ``// s1 is stack to hold nodes of first BST ` `    ``struct` `snode *s1 = NULL; ` ` `  `    ``// Current node of first BST ` `    ``struct` `node  *current1 = root1; ` ` `  `    ``// s2 is stack to hold nodes of second BST ` `    ``struct` `snode *s2 = NULL; ` ` `  `    ``// Current node of second BST ` `    ``struct` `node  *current2 = root2; ` ` `  `    ``// If first BST is empty, then output is inorder ` `    ``// traversal of second BST ` `    ``if` `(root1 == NULL) ` `    ``{ ` `        ``inorder(root2); ` `        ``return``; ` `    ``} ` `    ``// If second BST is empty, then output is inorder ` `    ``// traversal of first BST ` `    ``if` `(root2 == NULL) ` `    ``{ ` `        ``inorder(root1); ` `        ``return` `; ` `    ``} ` ` `  `    ``// Run the loop while there are nodes not yet printed. ` `    ``// The nodes may be in stack(explored, but not printed) ` `    ``// or may be not yet explored ` `    ``while` `(current1 != NULL || !isEmpty(s1) || ` `          ``current2 != NULL || !isEmpty(s2)) ` `    ``{ ` `        ``// Following steps follow iterative Inorder Traversal ` `        ``if` `(current1 != NULL || current2 != NULL ) ` `        ``{ ` `            ``// Reach the leftmost node of both BSTs and push ancestors of ` `            ``// leftmost nodes to stack s1 and s2 respectively ` `            ``if` `(current1 != NULL) ` `            ``{ ` `                ``push(&s1, current1); ` `                ``current1 = current1->left; ` `            ``} ` `            ``if` `(current2 != NULL) ` `            ``{ ` `                ``push(&s2, current2); ` `                ``current2 = current2->left; ` `            ``} ` ` `  `        ``} ` `        ``else` `        ``{ ` `            ``// If we reach a NULL node and either of the stacks is empty, ` `            ``// then one tree is exhausted, ptint the other tree ` `            ``if` `(isEmpty(s1)) ` `            ``{ ` `                ``while` `(!isEmpty(s2)) ` `                ``{ ` `                    ``current2 = pop (&s2); ` `                    ``current2->left = NULL; ` `                    ``inorder(current2); ` `                ``} ` `                ``return` `; ` `            ``} ` `            ``if` `(isEmpty(s2)) ` `            ``{ ` `                ``while` `(!isEmpty(s1)) ` `                ``{ ` `                    ``current1 = pop (&s1); ` `                    ``current1->left = NULL; ` `                    ``inorder(current1); ` `                ``} ` `                ``return` `; ` `            ``} ` ` `  `            ``// Pop an element from both stacks and compare the ` `            ``// popped elements ` `            ``current1 = pop(&s1); ` `            ``current2 = pop(&s2); ` ` `  `            ``// If element of first tree is smaller, then print it ` `            ``// and push the right subtree. If the element is larger, ` `            ``// then we push it back to the corresponding stack. ` `            ``if` `(current1->data < current2->data) ` `            ``{ ` `                ``printf``(``"%d "``, current1->data); ` `                ``current1 = current1->right; ` `                ``push(&s2, current2); ` `                ``current2 = NULL; ` `            ``} ` `            ``else` `            ``{ ` `                ``printf``(``"%d "``, current2->data); ` `                ``current2 = current2->right; ` `                ``push(&s1, current1); ` `                ``current1 = NULL; ` `            ``} ` `        ``} ` `    ``} ` `} ` ` `  `/* Driver program to test above functions */` `int` `main() ` `{ ` `    ``struct` `node  *root1 = NULL, *root2 = NULL; ` ` `  `    ``/* Let us create the following tree as first tree ` `            ``3 ` `          ``/  ` `         ``1    5 ` `     ``*/` `    ``root1 = newNode(3); ` `    ``root1->left = newNode(1); ` `    ``root1->right = newNode(5); ` ` `  `    ``/* Let us create the following tree as second tree ` `            ``4 ` `          ``/  ` `         ``2    6 ` `     ``*/` `    ``root2 = newNode(4); ` `    ``root2->left = newNode(2); ` `    ``root2->right = newNode(6); ` ` `  `    ``// Print sorted nodes of both trees ` `    ``merge(root1, root2); ` ` `  `    ``return` `0; ` `} `

Output:

```1 2 3 4 5 6
```

Time Complexity: O(m+n)
Auxiliary Space: O(height of the first tree + height of the second tree)