# Inorder Successor in Binary Search Tree

In Binary Tree, Inorder successor of a node is the next node in Inorder traversal of the Binary Tree. Inorder Successor is NULL for the last node in Inorder traversal.
In Binary Search Tree, Inorder Successor of an input node can also be defined as the node with the smallest key greater than the key of the input node. So, it is sometimes important to find next node in sorted order.

In the above diagram, inorder successor of 8 is 10, inorder successor of 10 is 12 and inorder successor of 14 is 20.

## Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

Method 1 (Uses Parent Pointer)
In this method, we assume that every node has a parent pointer.

The Algorithm is divided into two cases on the basis of the right subtree of the input node being empty or not.

Input: node, root // node is the node whose Inorder successor is needed.
Output: succ // succ is Inorder successor of node.

1. If right subtree of node is not NULL, then succ lies in right subtree. Do the following.
Go to right subtree and return the node with minimum key value in the right subtree.
2. If right sbtree of node is NULL, then succ is one of the ancestors. Do the following.
Travel up using the parent pointer until you see a node which is left child of its parent. The parent of such a node is the succ.

Implementation:
Note that the function to find InOrder Successor is highlighted (with gray background) in below code.

 `#include ` `#include ` ` `  `/* A binary tree node has data,  ` `   ``the pointer to left child ` `   ``and a pointer to right child */` `struct` `node { ` `    ``int` `data; ` `    ``struct` `node* left; ` `    ``struct` `node* right; ` `    ``struct` `node* parent; ` `}; ` ` `  `struct` `node* minValue(``struct` `node* node); ` ` `  `struct` `node* inOrderSuccessor( ` `    ``struct` `node* root, ` `    ``struct` `node* n) ` `{ ` `    ``// step 1 of the above algorithm ` `    ``if` `(n->right != NULL) ` `        ``return` `minValue(n->right); ` ` `  `    ``// step 2 of the above algorithm ` `    ``struct` `node* p = n->parent; ` `    ``while` `(p != NULL && n == p->right) { ` `        ``n = p; ` `        ``p = p->parent; ` `    ``} ` `    ``return` `p; ` `} ` ` `  `/* Given a non-empty binary search tree,  ` `    ``return the minimum data   ` `    ``value found in that tree. Note that  ` `    ``the entire tree does not need ` `    ``to be searched. */` `struct` `node* minValue(``struct` `node* node) ` `{ ` `    ``struct` `node* current = node; ` ` `  `    ``/* loop down to find the leftmost leaf */` `    ``while` `(current->left != NULL) { ` `        ``current = current->left; ` `    ``} ` `    ``return` `current; ` `} ` ` `  `/* Helper function that allocates a new  ` `    ``node with the given data and  ` `    ``NULL left and right pointers. */` `struct` `node* newNode(``int` `data) ` `{ ` `    ``struct` `node* node = (``struct` `node*) ` `        ``malloc``(``sizeof``( ` `            ``struct` `node)); ` `    ``node->data = data; ` `    ``node->left = NULL; ` `    ``node->right = NULL; ` `    ``node->parent = NULL; ` ` `  `    ``return` `(node); ` `} ` ` `  `/* Give a binary search tree and  ` `   ``a number, inserts a new node with     ` `    ``the given number in the correct  ` `    ``place in the tree. Returns the new ` `    ``root pointer which the caller should ` `    ``then use (the standard trick to  ` `    ``avoid using reference parameters). */` `struct` `node* insert(``struct` `node* node, ` `                    ``int` `data) ` `{ ` `    ``/* 1. If the tree is empty, return a new, ` `      ``single node */` `    ``if` `(node == NULL) ` `        ``return` `(newNode(data)); ` `    ``else` `{ ` `        ``struct` `node* temp; ` ` `  `        ``/* 2. Otherwise, recur down the tree */` `        ``if` `(data <= node->data) { ` `            ``temp = insert(node->left, data); ` `            ``node->left = temp; ` `            ``temp->parent = node; ` `        ``} ` `        ``else` `{ ` `            ``temp = insert(node->right, data); ` `            ``node->right = temp; ` `            ``temp->parent = node; ` `        ``} ` ` `  `        ``/* return the (unchanged) node pointer */` `        ``return` `node; ` `    ``} ` `} ` ` `  `/* Driver program to test above functions*/` `int` `main() ` `{ ` `    ``struct` `node *root = NULL, *temp, *succ, *min; ` ` `  `    ``// creating the tree given in the above diagram ` `    ``root = insert(root, 20); ` `    ``root = insert(root, 8); ` `    ``root = insert(root, 22); ` `    ``root = insert(root, 4); ` `    ``root = insert(root, 12); ` `    ``root = insert(root, 10); ` `    ``root = insert(root, 14); ` `    ``temp = root->left->right->right; ` ` `  `    ``succ = inOrderSuccessor(root, temp); ` `    ``if` `(succ != NULL) ` `        ``printf``( ` `            ``"\n Inorder Successor of %d is %d "``, ` `            ``temp->data, succ->data); ` `    ``else` `        ``printf``(``"\n Inorder Successor doesn't exit"``); ` ` `  `    ``getchar``(); ` `    ``return` `0; ` `} `

 `// Java program to find minimum ` `// value node in Binary Search Tree ` ` `  `// A binary tree node ` `class` `Node { ` ` `  `    ``int` `data; ` `    ``Node left, right, parent; ` ` `  `    ``Node(``int` `d) ` `    ``{ ` `        ``data = d; ` `        ``left = right = parent = ``null``; ` `    ``} ` `} ` ` `  `class` `BinaryTree { ` ` `  `    ``static` `Node head; ` ` `  `    ``/* Given a binary search tree and a number,  ` `     ``inserts a new node with the given number in  ` `     ``the correct place in the tree. Returns the new  ` `     ``root pointer which the caller should then use  ` `     ``(the standard trick to avoid using reference  ` `     ``parameters). */` `    ``Node insert(Node node, ``int` `data) ` `    ``{ ` ` `  `        ``/* 1. If the tree is empty, return a new,      ` `         ``single node */` `        ``if` `(node == ``null``) { ` `            ``return` `(``new` `Node(data)); ` `        ``} ` `        ``else` `{ ` ` `  `            ``Node temp = ``null``; ` ` `  `            ``/* 2. Otherwise, recur down the tree */` `            ``if` `(data <= node.data) { ` `                ``temp = insert(node.left, data); ` `                ``node.left = temp; ` `                ``temp.parent = node; ` `            ``} ` `            ``else` `{ ` `                ``temp = insert(node.right, data); ` `                ``node.right = temp; ` `                ``temp.parent = node; ` `            ``} ` ` `  `            ``/* return the (unchanged) node pointer */` `            ``return` `node; ` `        ``} ` `    ``} ` ` `  `    ``Node inOrderSuccessor(Node root, Node n) ` `    ``{ ` ` `  `        ``// step 1 of the above algorithm ` `        ``if` `(n.right != ``null``) { ` `            ``return` `minValue(n.right); ` `        ``} ` ` `  `        ``// step 2 of the above algorithm ` `        ``Node p = n.parent; ` `        ``while` `(p != ``null` `&& n == p.right) { ` `            ``n = p; ` `            ``p = p.parent; ` `        ``} ` `        ``return` `p; ` `    ``} ` ` `  `    ``/* Given a non-empty binary search  ` `       ``tree, return the minimum data   ` `       ``value found in that tree. Note that  ` `       ``the entire tree does not need ` `       ``to be searched. */` `    ``Node minValue(Node node) ` `    ``{ ` `        ``Node current = node; ` ` `  `        ``/* loop down to find the leftmost leaf */` `        ``while` `(current.left != ``null``) { ` `            ``current = current.left; ` `        ``} ` `        ``return` `current; ` `    ``} ` ` `  `    ``// Driver program to test above functions ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``BinaryTree tree = ``new` `BinaryTree(); ` `        ``Node root = ``null``, temp = ``null``, suc = ``null``, min = ``null``; ` `        ``root = tree.insert(root, ``20``); ` `        ``root = tree.insert(root, ``8``); ` `        ``root = tree.insert(root, ``22``); ` `        ``root = tree.insert(root, ``4``); ` `        ``root = tree.insert(root, ``12``); ` `        ``root = tree.insert(root, ``10``); ` `        ``root = tree.insert(root, ``14``); ` `        ``temp = root.left.right.right; ` `        ``suc = tree.inOrderSuccessor(root, temp); ` `        ``if` `(suc != ``null``) { ` `            ``System.out.println( ` `                ``"Inorder successor of "` `                ``+ temp.data + ``" is "` `+ suc.data); ` `        ``} ` `        ``else` `{ ` `            ``System.out.println( ` `                ``"Inorder successor does not exist"``); ` `        ``} ` `    ``} ` `} ` ` `  `// This code has been contributed by Mayank Jaiswal `

 `# Python program to find the inroder successor in a BST ` ` `  `# A binary tree node  ` `class` `Node: ` ` `  `    ``# Constructor to create a new node ` `    ``def` `__init__(``self``, key): ` `        ``self``.data ``=` `key  ` `        ``self``.left ``=` `None` `        ``self``.right ``=` `None` ` `  `def` `inOrderSuccessor(root, n): ` `     `  `    ``# Step 1 of the above algorithm ` `    ``if` `n.right ``is` `not` `None``: ` `        ``return` `minValue(n.right) ` ` `  `    ``# Step 2 of the above algorithm ` `    ``p ``=` `n.parent ` `    ``while``( p ``is` `not` `None``): ` `        ``if` `n !``=` `p.right : ` `            ``break`  `        ``n ``=` `p  ` `        ``p ``=` `p.parent ` `    ``return` `p ` ` `  `# Given a non-empty binary search tree, return the  ` `# minimum data value found in that tree. Note that the ` `# entire tree doesn't need to be searched ` `def` `minValue(node): ` `    ``current ``=` `node ` ` `  `    ``# loop down to find the leftmost leaf ` `    ``while``(current ``is` `not` `None``): ` `        ``if` `current.left ``is` `None``: ` `            ``break` `        ``current ``=` `current.left ` ` `  `    ``return` `current ` ` `  ` `  `# Given a binary search tree and a number, inserts a ` `# new node with the given number in the correct place ` `# in the tree. Returns the new root pointer which the ` `# caller should then use( the standard trick to avoid  ` `# using reference parameters) ` `def` `insert( node, data): ` ` `  `    ``# 1) If tree is empty then return a new singly node ` `    ``if` `node ``is` `None``: ` `        ``return` `Node(data) ` `    ``else``: ` `        `  `        ``# 2) Otherwise, recur down the tree ` `        ``if` `data <``=` `node.data: ` `            ``temp ``=` `insert(node.left, data) ` `            ``node.left ``=` `temp  ` `            ``temp.parent ``=` `node ` `        ``else``: ` `            ``temp ``=` `insert(node.right, data) ` `            ``node.right ``=` `temp  ` `            ``temp.parent ``=` `node ` `         `  `        ``# return  the unchanged node pointer ` `        ``return` `node ` ` `  ` `  `# Driver progam to test above function ` ` `  `root ``=` `None` ` `  `# Creating the tree given in the above diagram  ` `root ``=` `insert(root, ``20``) ` `root ``=` `insert(root, ``8``); ` `root ``=` `insert(root, ``22``); ` `root ``=` `insert(root, ``4``); ` `root ``=` `insert(root, ``12``); ` `root ``=` `insert(root, ``10``);   ` `root ``=` `insert(root, ``14``);     ` `temp ``=` `root.left.right.right  ` ` `  `succ ``=` `inOrderSuccessor( root, temp) ` `if` `succ ``is` `not` `None``: ` `    ``print` `"\nInorder Successor of % d is % d "` `\ ` `            ``%``(temp.data, succ.data) ` `else``: ` `    ``print` `"\nInorder Successor doesn't exist"` ` `  `# This code is contributed by Nikhil Kumar Singh(nickzuck_007) `

Output:
`Inorder Successor of 14 is 20`

Complexity Analysis:

• Time Complexity: O(h), where h is the height of the tree.
As in the second case(suppose skewed tree) we have to travel all the way towards the root.
• Auxiliary Space: O(1).
Due to no use of any data structure for storing values.

Method 2 (Search from root)
Parent pointer is NOT needed in this algorithm. The Algorithm is divided into two cases on the basis of right subtree of the input node being empty or not.

Input: node, root // node is the node whose Inorder successor is needed.
Output: succ // succ is Inorder successor of node.

1. If right subtree of node is not NULL, then succ lies in right subtree. Do the following.
Go to right subtree and return the node with minimum key value in the right subtree.
2. If right subtree of node is NULL, then start from the root and us search like technique. Do the following.
Travel down the tree, if a node’s data is greater than root’s data then go right side, otherwise, go to left side.
 `struct` `node* inOrderSuccessor( ` `    ``struct` `node* root, ` `    ``struct` `node* n) ` `{ ` `    ``// step 1 of the above algorithm ` `    ``if` `(n->right != NULL) ` `        ``return` `minValue(n->right); ` ` `  `    ``struct` `node* succ = NULL; ` ` `  `    ``// Start from root and search for ` `    ``// successor down the tree ` `    ``while` `(root != NULL) { ` `        ``if` `(n->data < root->data) { ` `            ``succ = root; ` `            ``root = root->left; ` `        ``} ` `        ``else` `if` `(n->data > root->data) ` `            ``root = root->right; ` `        ``else` `            ``break``; ` `    ``} ` ` `  `    ``return` `succ; ` `} `

Thanks to R.Srinivasan for suggesting this method.

Complexity Analysis:

• Time Complexity: O(h), where h is the height of the tree.
In the worst case as explained above we travel the whole height of the tree
• Auxiliary Space: O(1).
Due to no use of any data structure for storing values.