**Exercise 8.** In a red-black tree which is initially empty, we insert in order nodes with keys 1, 2, 3, 4. Draw the tree after these insertions (single-circled nodes are black, and double-circled nodes are red).### Explanation:This task involves maintaining a balanced red-black tree structure while inserting the nodes with keys 1, 2, 3, and 4 in that order. A red-black tree is a type of self-balancing binary search tree. It has the following properties:1. Each node is either red or black.2. The root is black.3. All leaves (NIL) are black.4. If a node is red, then both its children are black.5. Every path from a node to its descendant NIL nodes has the same number of black nodes.### Tree Diagram:The diagram provided in the image depicts the structure of the red-black tree during and after each insertion as follows:- After inserting key `1`, the tree consists of a single black node labeled `1`.- Upon inserting key `2`, it becomes the right child of node `1`, with node `2` being red.- Inserting key `3` causes a rotation to maintain tree properties, making node `2` the black root, with `1` as its left child (red), and `3` as its right child (red).- Inserting key `4` leads to a further restructuring, ensuring that node `2` remains the root, node `1` is its left black child, node `3` becomes the right black child of `2`, and node `4` is the red right child of `3`.This process preserves the properties of a red-black tree after each insertion. Each node is visually distinguished with single and double circles to denote black and red colors, respectively.

Answered: Exercise 8. In a red-black tree which…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

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**Exercise 8.** In a red-black tree which is initially empty, we insert in order nodes with keys 1, 2, 3, 4. Draw the tree after these insertions (single-circled nodes are black, and double-circled nodes are red).

### Explanation:

This task involves maintaining a balanced red-black tree structure while inserting the nodes with keys 1, 2, 3, and 4 in that order. A red-black tree is a type of self-balancing binary search tree. It has the following properties:

1. Each node is either red or black.
2. The root is black.
3. All leaves (NIL) are black.
4. If a node is red, then both its children are black.
5. Every path from a node to its descendant NIL nodes has the same number of black nodes.

### Tree Diagram:

The diagram provided in the image depicts the structure of the red-black tree during and after each insertion as follows:

- After inserting key `1`, the tree consists of a single black node labeled `1`.
- Upon inserting key `2`, it becomes the right child of node `1`, with node `2` being red.
- Inserting key `3` causes a rotation to maintain tree properties, making node `2` the black root, with `1` as its left child (red), and `3` as its right child (red).
- Inserting key `4` leads to a further restructuring, ensuring that node `2` remains the root, node `1` is its left black child, node `3` becomes the right black child of `2`, and node `4` is the red right child of `3`.

This process preserves the properties of a red-black tree after each insertion. Each node is visually distinguished with single and double circles to denote black and red colors, respectively.

Expert Solution

Step 1

following are every step to insert a node in the red-black tree.

A red-black tree is a kind of self-balancing binary search tree where each node has an extra bit, and that bit is often interpreted as the color (red or black).

These colors are used to ensure that the tree remains balanced during insertions and deletions.

Rule: if the parent node is red then its children are black.

Rule: if the parent node is black then the child can be red or black.

so the following are the representation so red nodes represent red and simple gray nodes are black nodes.

initially, the tree is empty:

1. insert 1

2. insert 2

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