For the following list :
4
6
,

4
3
,

5
5
,

3
4
,

2
4
,

2
5
,

6
0
,

9
1
,

9
8
,

7
0
,

3
9
,

4
1
,

9
2
,

6
4
,

8
2

a
.
construct
(
draw
)
the
(
2
,
4
)
-
tree
b
.
construct
(
draw
)
the red
-
black tree
(
by converting the tree from
(
a
)

)

inserting their elements successively, starting with the empty tree.

Question

﻿For the following list :
4
6
,
 
4
3
,
 
5
5
,
 
3
4
,
 
2
4
,
 
2
5
,
 
6
0
,
 
9
1
,
 
9
8
,
 
7
0
,
 
3
9
,
 
4
1
,
 
9
2
,
 
6
4
,
 
8
2
 ﻿
a
.
 ﻿construct 
(
draw
)
 ﻿the 
(
2
,
4
)
-
tree
b
.
 ﻿construct 
(
draw
)
 ﻿the red
-
black tree 
(
 ﻿by converting the tree from 
(
a 
)
 
 
)
 ﻿
inserting their elements successively, starting with the empty tree.

Accepted Answer

∻Step-by-step Instruction∻ ‖ ▪ ⋮ Define the main functions required here: construct a 2-4 tree and a red-black tree. ‖ ‖ ▪ ⋮ Choose to represent the trees visually and insert elements successively. ‖ ‖ ▪ ⋮ Define concrete use cases and testing cases for the function: when user inserts elements, the tree should maintain its properties. ‖ ### Constructing a 2-4 Tree ∻Explanation∻ ⚹A 2-4 tree is a self-balancing tree where every node has either 2, 3, or 4 children.⚹ 1. **Initialize the tree**: Start with an empty tree. 2. **Insert elements**: Insert each element from the list into the tree while maintaining the 2-4 tree properties. 3. **Split nodes**: If a node exceeds 4 children, split it into two nodes and propagate the middle element up. ### Constructing a Red-Black Tree ∻Explanation∻ ⚹A red-black tree is a binary search tree with an extra bit for denoting the color of the node, either red or black.⚹ 1. **Initialize the tree**: Start with an empty tree. 2. **Insert elements**: Insert each element from the list into the tree while maintaining the red-black tree properties. 3. **Recolor and rotate**: After each insertion, recolor and rotate the tree to maintain balance. ### Example Code for 2-4 Tree ```python class Node: def __init__(self, keys=None, children=None): self.keys = keys or [] self.children = children or [] class TwoFourTree: def __init__(self): self.root = Node() def insert(self, key): # Insert key into the 2-4 tree pass # Implementation of insertion logic def split(self, node): # Split a node if it has more than 4 children pass # Implementation of split logic def display(self): # Display the tree structure pass # Implementation of display logic ``` ### Example Code for Red-Black Tree ```python class RedBlackNode: def __init__(self, key, color='red', left=None, right=None, parent=None): self.key = key self.color = color self.left = left self.right = right self.parent = parent class RedBlackTree: def __init__(self): self.NIL = RedBlackNode(key=None, color='black') self.root = self.NIL def insert(self, key): # Insert key into the red-black tree pass # Implementation of insertion logic def rotate_left(self, node): # Rotate node to the left pass # Implementation of left rotation def rotate_right(self, node): # Rotate node to the right pass # Implementation of right rotation def fix_insert(self, node): # Fix the tree after insertion pass # Implementation of fix logic def display(self): # Display the tree structure pass # Implementation of display logic ``` ### Main Function for Testing ```python def main(): # Test 2-4 Tree two_four_tree = TwoFourTree() elements = [(4, 6), (4, 3), (5, 5), (3, 4), (2, 4), (2, 5), (6, 0), (9, 1), (9, 8), (7, 0), (3, 9), (4, 1), (9, 2), (6, 4), (8, 2)] for element in elements: two_four_tree.insert(element) two_four_tree.display() # Test Red-Black Tree red_black_tree = RedBlackTree() for element in elements: red_black_tree.insert(element) red_black_tree.display() if __name__ == "__main__": main() ``` ∻Time Complexity∻ ⚹The time complexity for insertion in both 2-4 trees and red-black trees is $O(\log n)$.⚹ ∻Space Complexity∻ ⚹The space complexity for both trees is $O(n)$, where $n$ is the number of elements in the tree.⚹

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