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.

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.

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.
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.

C

#include <stdio.h>
#include <stdlib.h>
 
/* 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

// 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

# 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.

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.
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.

Below is the implementation of the above approach:

C

// C program for above approach
#include <stdio.h>
#include <stdlib.h>
 
/* 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);
 
    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;
}
 
/* 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;
     
    // Function Call
    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;
}
 
// Thanks to R.Srinivasan for suggesting this method.

Python3

# 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
    succ=Node(None)
     
     
    while( root):
        if(root.data<n.data):
            root=root.right
        elif(root.data>n.data):
            succ=root
            root=root.left
        else:
            break
    return succ
 
# 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
if __name__ == "__main__":
    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 
 
  succ = inOrderSuccessor( root, temp)
  if succ is not None:
      print("Inorder Successor of" ,
               temp.data ,"is" ,succ.data)
  else:
      print("InInorder Successor doesn't exist")

Output

 Inorder Successor of 14 is 20

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.

https://www.youtube.com/watch?v=kr3BOCNEYHI

References:
http://net.pku.edu.cn/~course/cs101/2007/resource/Intro2Algorithm/book6/chap13.htm
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