Lecture on Deleting a Node in a Binary Search Tree (BST)

Introduction

Discussion on Alfred Nobel Prize for Science
Reference to company partnerships and job platforms

Key Concepts

Binary Search Tree (BST)

Definition: A tree structure where each node has at most two children, left child < node < right child.
Application: Efficient searching, inserting, and deletion of nodes.

Deleting a Node in BST

Identification of Node: Locate the node to be deleted.
Three Possible Cases:
1. Node is a Leaf: Simply remove the node.
2. Node has One Child: Bypass the node by linking its parent to its child directly.
3. Node has Two Children: Find the in-order successor (smallest node in the right subtree) or in-order predecessor (largest node in the left subtree), replace the node with the successor/predecessor, and delete the successor/predecessor.

Detailed Steps

Case 1: Node is a Leaf

Remove the node without any further modifications.
Example: If node 10 is a leaf, simply delete node 10.

Case 2: Node has One Child

Link the child directly to the parent of the node to be deleted.
Example: If node 10 has a single child node 15, link node 15 directly to the parent of node 10.

Case 3: Node has Two Children

Find Successor/Predecessor: Locate the in-order successor or predecessor.
- In-order Successor: Smallest node in the right subtree.
- In-order Predecessor: Largest node in the left subtree.
Replace Node: Replace the node to be deleted with its successor/predecessor.
Delete Successor/Predecessor: Remove the successor or predecessor node from its original position.

Complexity and Edge Cases

Time Complexity: Generally, O(log n) for balanced trees, but can degrade to O(n) for unbalanced trees.
Edge Cases: Special attention needed for unique scenarios like duplicate nodes, and balancing trees post-deletion.

Practical Implementation

Pseudocode for Node Deletion

function deleteNode(root, key):
    if root is None:
        return root
    if key < root.value:
        root.left = deleteNode(root.left, key)
    elif key > root.value:
        root.right = deleteNode(root.right, key)
    else:
        if root.left is None:
            return root.right
        elif root.right is None:
            return root.left
        temp = minValueNode(root.right)
        root.value = temp.value
        root.right = deleteNode(root.right, temp.value)
    return root

Example Execution

Example provided to illustrate each case:
- Leaf Node Deletion
- Node with One Child Deletion
- Node with Two Children Deletion

Summary

Effective management of BST requires understanding and proper handling of deletions to maintain tree properties.
Importance of ensuring minimum disruptions to tree structure during deletions.

Supplementary Materials

Further readings and code examples.
Video walkthroughs and explanations.

Conclusion

Deleting nodes in BST is foundational for maintaining efficient tree operations.
Emphasis on practice and deeper exploration of edge cases for robust understanding.