Given a binary search tree and a **node** of the binary search tree, the task is to delete the node from the Binary Search tree Iteratiely.

Here are the three cases that arise while performing a delete operation on a BST:

**1. Case 1:** Node to be deleted is a leaf node. Directly delete the node from the tree.

10 10 / \ delete(5) / \ 7 15 ---------> 7 15 / \ / \ \ / \ 5 8 11 18 8 11 18

**2. Case 2:** Node to be deleted is an internal node with two children. Copy the contents of the inorder successor of the node to be deleted and delete the inorder successor. The inorder successor can be found by finding the minimum element in the right subtree of the node.

inorderSuccessor(10) = 11.

10 11 / \ delete(10) / \ 7 15 ---------> 7 15 / \ / \ / \ \ 5 8 11 18 5 8 18

**3. Case 3:** Node to be deleted is an internal node with one child. For this case, delete the node and move its child up to take its place.

10 10 / \ delete(15) / \ 7 15 ---------> 7 11 / \ / / \ 5 8 11 5 8

The intuition behind deleting the inorder successor in Case 2 is that the inorder successor of a node with two children will always be greater than all elements in the left sub-tree of the of the node since it is the smallest node in the right sub-tree of the node and the inorder successor of the node will always be smaller than all other nodes in the right sub-tree of the node.

This preserves the BST property of all nodes in the left sub-tree of a given node are smaller than the given node and all nodes in the right sub-tree of the given node are greater than the given node.

Below is the implementation of the above approach:

## C++

`// C++ implementation to delete` `// a node in the BST` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Structure of the node` `typedef` `struct` `treeNode {` ` ` `int` `data;` ` ` `struct` `treeNode* left;` ` ` `struct` `treeNode* right;` `} treeNode;` `// Utility function to print` `// the inorder traversal of the BST.` `void` `inorder(treeNode* root)` `{` ` ` `if` `(root != NULL) {` ` ` `inorder(root->left);` ` ` `cout << root->data << ` `' '` `;` ` ` `inorder(root->right);` ` ` `}` `}` `// Utility function to insert` `// nodes into our BST` `treeNode* insert(treeNode* root, ` `int` `key)` `{` ` ` `// Check if tree is empty` ` ` `if` `(root == NULL) {` ` ` `treeNode* temp;` ` ` `temp = (treeNode*)` `malloc` `(` `sizeof` `(treeNode));` ` ` `temp->data = key;` ` ` `temp->left = NULL;` ` ` `temp->right = NULL;` ` ` `return` `temp;` ` ` `}` ` ` `if` `(key < root->data) {` ` ` `// if the key to be inserted` ` ` `// is lesser than the root,` ` ` `// insert into the left subtree,` ` ` `// and recursively call` ` ` `// the insert function with the` ` ` `// root->left as the new root.` ` ` `root->left = insert(root->left, key);` ` ` `}` ` ` `else` `{` ` ` `// if the key to be inserted` ` ` `// is greater than the root,` ` ` `// insert into the right subtree,` ` ` `// and recursively call` ` ` `// the insert function with the` ` ` `// root->right as the new root.` ` ` `root->right = insert(root->right, key);` ` ` `}` ` ` `return` `root;` `}` `// Iterative Function to delete` `// 'key' from the BST.` `treeNode* deleteIterative(treeNode* root, ` `int` `key)` `{` ` ` `treeNode* curr = root;` ` ` `treeNode* prev = NULL;` ` ` `// Check if the key is actually` ` ` `// present in the BST.` ` ` `// the variable prev points to` ` ` `// the parent of the key to be deleted.` ` ` `while` `(curr != NULL && curr->data != key) {` ` ` `prev = curr;` ` ` `if` `(key < curr->data)` ` ` `curr = curr->left;` ` ` `else` ` ` `curr = curr->right;` ` ` `}` ` ` `if` `(curr == NULL) {` ` ` `cout << ` `"Key "` `<< key << ` `" not found in the"` ` ` `<< ` `" provided BST.\n"` `;` ` ` `return` `root;` ` ` `}` ` ` `// Check if the node to be` ` ` `// deleted has atmost one child.` ` ` `if` `(curr->left == NULL || curr->right == NULL) {` ` ` `// newCurr will replace` ` ` `// the node to be deleted.` ` ` `treeNode* newCurr;` ` ` `// if the left child does not exist.` ` ` `if` `(curr->left == NULL)` ` ` `newCurr = curr->right;` ` ` `else` ` ` `newCurr = curr->left;` ` ` `// check if the node to` ` ` `// be deleted is the root.` ` ` `if` `(prev == NULL)` ` ` `return` `newCurr;` ` ` `// check if the node to be deleted` ` ` `// is prev's left or right child` ` ` `// and then replace this with newCurr` ` ` `if` `(curr == prev->left)` ` ` `prev->left = newCurr;` ` ` `else` ` ` `prev->right = newCurr;` ` ` `// free memory of the` ` ` `// node to be deleted.` ` ` `free` `(curr);` ` ` `}` ` ` `// node to be deleted has` ` ` `// two children.` ` ` `else` `{` ` ` `treeNode* p = NULL;` ` ` `treeNode* temp;` ` ` `// Compute the inorder successor` ` ` `temp = curr->right;` ` ` `while` `(temp->left != NULL) {` ` ` `p = temp;` ` ` `temp = temp->left;` ` ` `}` ` ` `// check if the parent of the inorder` ` ` `// successor is the curr or not(i.e. curr=` ` ` `// the node which has the same data as` ` ` `// the given data by the user to be` ` ` `// deleted). if it isn't, then make the` ` ` `// the left child of its parent equal to` ` ` `// the inorder successor'd right child.` ` ` `if` `(p != NULL)` ` ` `p->left = temp->right;` ` ` `// if the inorder successor was the` ` ` `// curr (i.e. curr = the node which has the` ` ` `// same data as the given data by the` ` ` `// user to be deleted), then make the` ` ` `// right child of the node to be` ` ` `// deleted equal to the right child of` ` ` `// the inorder successor.` ` ` `else` ` ` `curr->right = temp->right;` ` ` `curr->data = temp->data;` ` ` `free` `(temp);` ` ` `}` ` ` `return` `root;` `}` `// Driver Code` `int` `main()` `{` ` ` `/*` ` ` `10` ` ` `/ \` ` ` `7 15` ` ` `/ \ / \` ` ` `5 8 11 18` ` ` `*/` ` ` `treeNode* root = NULL;` ` ` `root = insert(root, 10);` ` ` `root = insert(root, 7);` ` ` `root = insert(root, 5);` ` ` `root = insert(root, 8);` ` ` `root = insert(root, 15);` ` ` `root = insert(root, 11);` ` ` `root = insert(root, 18);` ` ` `cout << ` `"Inorder traversal "` ` ` `<< ` `"of original BST:\n"` `;` ` ` `inorder(root);` ` ` `cout << ` `'\n'` `;` ` ` `// delete node with data value 11 (leaf)` ` ` `root = deleteIterative(root, 11);` ` ` `cout << ` `"\nDeletion of 11\n"` `;` ` ` `cout << ` `"Inorder traversal post deletion:\n"` `;` ` ` `inorder(root);` ` ` `cout << ` `'\n'` `;` ` ` `// delete node with data value 15` ` ` `// (internal node with one child)` ` ` `root = deleteIterative(root, 15);` ` ` `cout << ` `"\nDeletion of 15\n"` `;` ` ` `cout << ` `"Inorder traversal post deletion:\n"` `;` ` ` `inorder(root);` ` ` `cout << ` `'\n'` `;` ` ` `// delete node with data value 10` ` ` `// (root, two children)` ` ` `root = deleteIterative(root, 10);` ` ` `cout << ` `"\nDeletion of 10\n"` `;` ` ` `cout << ` `"Inorder traversal post deletion:\n"` `;` ` ` `inorder(root);` ` ` `cout << ` `'\n'` `;` ` ` `return` `0;` `}` |

## Python3

`# Python implementation to delete` `# a node in the Binary Search Tree` `# Class for a node of BST.` `class` `Node:` ` ` `def` `__init__(` `self` `, data):` ` ` `self` `.data ` `=` `data` ` ` `self` `.left ` `=` `None` ` ` `self` `.right ` `=` `None` `# Utility function to print` `# the inorder traversal of the BST` `def` `inorder(root):` ` ` `if` `root !` `=` `None` `:` ` ` `inorder(root.left)` ` ` `print` `(root.data, end` `=` `" "` `)` ` ` `inorder(root.right)` `# Utility function to insert` `# nodes into our BST` `def` `insert(root, key):` ` ` `# check if tree is empty` ` ` `if` `root ` `=` `=` `None` `:` ` ` `temp ` `=` `Node(key)` ` ` `return` `temp` ` ` `if` `key < root.data:` ` ` `"""` ` ` `if the key to be inserted is` ` ` `lesser than the root,` ` ` `insert into the left subtree,` ` ` `and recursively call` ` ` `the insert function with` ` ` `the root.left as the new root.` ` ` `"""` ` ` `root.left ` `=` `insert(root.left, key)` ` ` `else` `:` ` ` `"""` ` ` `if the key to be inserted is` ` ` `greater than the root,` ` ` `insert into the right subtree,` ` ` `and recursively call` ` ` `the insert function with the` ` ` `root->right as the new root.` ` ` `"""` ` ` `root.right ` `=` `insert(root.right, key)` ` ` `return` `root` `# Iterative appraoch to` `# delete 'key' from the BST.` `def` `deleteIterative(root, key):` ` ` `curr ` `=` `root` ` ` `prev ` `=` `None` ` ` `# First check if the key is` ` ` `# actually present in the BST.` ` ` `# the variable prev points to the` ` ` `# parent of the key to be deleted` ` ` `while` `(curr !` `=` `None` `and` `curr.data !` `=` `key):` ` ` `prev ` `=` `curr` ` ` `if` `curr.data < key:` ` ` `curr ` `=` `curr.right` ` ` `else` `:` ` ` `curr ` `=` `curr.left` ` ` `if` `curr ` `=` `=` `None` `:` ` ` `print` `("Key ` `%` `d ` `not` `found ` `in` `\` ` ` `the provided BST." ` `%` `key)` ` ` `return` `root` ` ` `# Check if the node to be` ` ` `# deleted has atmost one child` ` ` `if` `curr.left ` `=` `=` `None` `or` `\` ` ` `curr.right ` `=` `=` `None` `:` ` ` `# newCurr will replace` ` ` `# the node to be deleted.` ` ` `newCurr ` `=` `None` ` ` `# if the left child does not exist.` ` ` `if` `curr.left ` `=` `=` `None` `:` ` ` `newCurr ` `=` `curr.right` ` ` `else` `:` ` ` `newCurr ` `=` `curr.left` ` ` `# check if the node to` ` ` `# be deleted is the root.` ` ` `if` `prev ` `=` `=` `None` `:` ` ` `return` `newCurr` ` ` `# Check if the node to be` ` ` `# deleted is prev's left or` ` ` `# right child and then` ` ` `# replace this with newCurr` ` ` `if` `curr ` `=` `=` `prev.left:` ` ` `prev.left ` `=` `newCurr` ` ` `else` `:` ` ` `prev.right ` `=` `newCurr` ` ` `curr ` `=` `None` ` ` `# node to be deleted` ` ` `# has two children.` ` ` `else` `:` ` ` `p ` `=` `None` ` ` `temp ` `=` `None` ` ` `# Compute the inorder` ` ` `# successor of curr.` ` ` `temp ` `=` `curr.right` ` ` `while` `(temp.left !` `=` `None` `):` ` ` `p ` `=` `temp` ` ` `temp ` `=` `temp.left` ` ` `# check if the parent of the` ` ` `# inorder successor is the root or not.` ` ` `# if it isn't, then make the left` ` ` `# child of its parent equal to the` ` ` `# inorder successor's right child.` ` ` `if` `p !` `=` `None` `:` ` ` `p.left ` `=` `temp.right` ` ` `else` `:` ` ` `# if the inorder successor was` ` ` `# the root, then make the right child` ` ` `# of the node to be deleted equal` ` ` `# to the right child of the inorder` ` ` `# successor.` ` ` `curr.right ` `=` `temp.right` ` ` `curr.data ` `=` `temp.data` ` ` `temp ` `=` `None` ` ` `return` `root` `# Function to create the BST` `# and call the Delete Function` `def` `main():` ` ` `"""` ` ` `10 ` ` ` `/ \ ` ` ` `7 15 ` ` ` `/ \ / \ ` ` ` `5 8 11 18 ` ` ` `"""` ` ` `root ` `=` `None` ` ` `root ` `=` `insert(root, ` `10` `)` ` ` `root ` `=` `insert(root, ` `7` `)` ` ` `root ` `=` `insert(root, ` `5` `)` ` ` `root ` `=` `insert(root, ` `8` `)` ` ` `root ` `=` `insert(root, ` `15` `)` ` ` `root ` `=` `insert(root, ` `11` `)` ` ` `root ` `=` `insert(root, ` `18` `)` ` ` `print` `(` `"Inorder traversal of original BST:"` `)` ` ` `inorder(root)` ` ` `print` `(` `"\n"` `)` ` ` `# delete node with data value 11 (leaf)` ` ` `root ` `=` `deleteIterative(root, ` `11` `)` ` ` `print` `(` `"Deletion of 11"` `)` ` ` `print` `(` `"Inorder traversal post deletion:"` `)` ` ` `inorder(root)` ` ` `print` `(` `"\n"` `)` ` ` `# delete node with data value 15` ` ` `# (internal node with one child)` ` ` `root ` `=` `deleteIterative(root, ` `15` `)` ` ` `print` `(` `"Deletion of 15"` `)` ` ` `print` `(` `"Inorder traversal post deletion:"` `)` ` ` `inorder(root)` ` ` `print` `(` `"\n"` `)` ` ` `# delete node with data value 10` ` ` `# (root, two children)` ` ` `root ` `=` `deleteIterative(root, ` `10` `)` ` ` `print` `(` `"Deletion of 10"` `)` ` ` `print` `(` `"Inorder traversal post deletion:"` `)` ` ` `inorder(root)` ` ` `print` `()` `# Driver Code` `if` `__name__ ` `=` `=` `"__main__"` `:` ` ` `main()` |

**Output:**

Inorder traversal of original BST: 5 7 8 10 11 15 18 Deletion of 11 Inorder traversal post deletion: 5 7 8 10 15 18 Deletion of 15 Inorder traversal post deletion: 5 7 8 10 18 Deletion of 10 Inorder traversal post deletion: 5 7 8 18

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