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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 nodeclass 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 = Nonedef inOrderSuccessor(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 searcheddef 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 functionroot = 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) |
C#
// C# program to find minimum// value node in Binary Search Treeusing System;// A binary tree nodepublic class Node { public int data; public Node left, right, parent; public Node(int d) { data = d; left = right = parent = null; } }public 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) { Console.WriteLine( "Inorder successor of " + temp.data + " is " + suc.data); } else { Console.WriteLine( "Inorder successor does not exist"); } }}// This code is contributed by aashish1995 |
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 use 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 = Nonedef 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 searcheddef 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 functionif __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://youtu.be/kr3BOCNEYHI?list=PLqM7alHXFySHCXD7r1J0ky9Zg_GBB1dbk
References:
http://net.pku.edu.cn/~course/cs101/2007/resource/Intro2Algorithm/book6/chap13.htm
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
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