Lecture Notes: Deletion in Red-Black Trees

Introduction

• Topic: Deletion and Red-Black Trees
• Reminder: Watch the previous videos in the playlist for foundational knowledge.

Review of Red-Black Trees

• Each node is either red or black.
• The root and leaves (nil nodes) are black.
• If a node is red, its children are black.
• All paths from a node to its null descendants must contain the same number of black nodes.

Key Sections of the Lecture

1. Transplant Method
2. Delete Method
3. Delete Fix-Up Method

Transplant Method

• Purpose: Moves subtrees within the red-black tree.
• Code Overview:
  • If U is the root:
    • Set root equal to V.
    • V's parent is set to U's parent (none).
  • If U is the left child:
    • Parent's left child is set to V.
    • V's parent is set to U's parent.
  • If U is the right child:
    • Parent's right child is set to V.
    • V's parent is set to U's parent.
• Note: Updating V's children is the responsibility of the calling function.

Delete Method

• Grouped into three cases:
  1. When the node to delete has a nil left child.
  2. When the right child is nil.
  3. When neither child is nil.

Case 1: Nil Left Child

• Example: Deleting node 19 (let's call it Z).
  • Note Z's original color (black).
  • Call transplant with Z and its right child X.
  • Original color was black; call deleteFixUp with node X.

Case 2: Nil Right Child

• Example: Adding node 1 and deleting node 5.
  • Note Z's original color and call transplant with Z and its left child X.
  • Call deleteFixUp with X because original color was black.

Case 3: Neither Child is Nil

• Example: Deleting node 12 (Z).
  • Find minimum in Z's right child's subtree (node 13, Y).
  • Note Y's original color (black) and label Y's right child as X.
  • Call transplant with Y and X.
  • Change pointers to set Y's right child to 15 and set 15's parent to Y.
  • Call transplant again with Z and Y.
  • Fix pointers to replace Z with Y.
  • Final step: Call deleteFixUp with X to ensure proper coloring.

Conclusion

• Covered:
  • Transplant method.
  • Three cases for delete operation.
• Upcoming content: Delete fix-up method in the next video.
• Additional resources: Code linked in the description for practical understanding.
• Thank you for watching!