I would like to upgrade the RB tree shown below by inserting the number 161 into it. Show how this can be done such that after the insertion it remains to be a valid RB tree. For full credit, you should show intermediate steps (i.e., by showing how the tree is being transformed into the final one) by drawing a few trees.

Database System Concepts
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ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
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Chapter1: Introduction
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### RB Tree Insertion Exercise

**Problem Statement:**

I would like to upgrade the RB tree shown below by inserting the number 161 into it. Show how this can be done such that after the insertion it remains a valid RB tree. For full credit, you should show intermediate steps (i.e., by showing how the tree is being transformed into the final one) by drawing a few trees.

**RB Tree Diagram:**

- **Nodes:** 
  - **Root node (42):** Black
  - **Left subtree:**
    - Node (5): Red
      - Left child (2): Black
        - Left child (1): Red
      - Right child (10): Black
  - **Right subtree:**
    - Node (150): Black
      - Right child (200): Red

**Explanation of the Tree Structure:**

1. **Root Node (42)**: This is a black node and forms the top of the tree.
2. **Left Subtree**:
   - The red node (5) is the left child of the root node.
   - Node (2) is the left child of (5) and is black.
   - Node (1) is the left child of (2) and is red.
   - Node (10) is the right child of (5) and is black.
3. **Right Subtree**:
   - The black node (150) is the right child of the root node.
   - Node (200) is the right child of (150) and is red.

**Task:**

To insert 161, ensure that the rules of red-black trees are maintained:
- Every node is either red or black.
- The root is black.
- Red nodes cannot have red children.
- Every path from a node to its descendant null nodes has the same number of black nodes.

For a valid transformation, remember to address violations of these rules through rotations and color flips. Intermediate steps should focus on maintaining these properties as you insert 161 into the appropriate position.
Transcribed Image Text:### RB Tree Insertion Exercise **Problem Statement:** I would like to upgrade the RB tree shown below by inserting the number 161 into it. Show how this can be done such that after the insertion it remains a valid RB tree. For full credit, you should show intermediate steps (i.e., by showing how the tree is being transformed into the final one) by drawing a few trees. **RB Tree Diagram:** - **Nodes:** - **Root node (42):** Black - **Left subtree:** - Node (5): Red - Left child (2): Black - Left child (1): Red - Right child (10): Black - **Right subtree:** - Node (150): Black - Right child (200): Red **Explanation of the Tree Structure:** 1. **Root Node (42)**: This is a black node and forms the top of the tree. 2. **Left Subtree**: - The red node (5) is the left child of the root node. - Node (2) is the left child of (5) and is black. - Node (1) is the left child of (2) and is red. - Node (10) is the right child of (5) and is black. 3. **Right Subtree**: - The black node (150) is the right child of the root node. - Node (200) is the right child of (150) and is red. **Task:** To insert 161, ensure that the rules of red-black trees are maintained: - Every node is either red or black. - The root is black. - Red nodes cannot have red children. - Every path from a node to its descendant null nodes has the same number of black nodes. For a valid transformation, remember to address violations of these rules through rotations and color flips. Intermediate steps should focus on maintaining these properties as you insert 161 into the appropriate position.
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