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The correct Huffman tree :: 120 0 306 (186)

The correct Huffman tree :: 120 0 306 (186)

Database System Concepts
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
Problem 1PE
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Hello. I have the problem and its answer. I am trying to understand how the tree was built, but I get really confused and lost from node 65 to upper part of the tree. I do not understand how 37, 42, 42 were added. Please provided as much detail as you can. Thank you! 

# The Correct Huffman Tree This diagram illustrates a Huffman tree, which is a binary tree used for data compression. Each node in the tree represents a cumulative frequency, while the leaves encase characters and their respective frequencies. ### Explanation of the Tree: 1. **Root Node (306)**: The topmost node of the tree holds the total frequency of all characters combined, which is 306. 2. **First Level**: - **Left Child (120)**: This node branches out directly from the root and represents a frequency of 120 for character 'E'. - **Right Child (186)**: This node leads to further branches with a cumulative frequency of 186. 3. **Second Level**: - From node 186: - **Left Child (79)**: - **Left Sub-child (37)**: Represents the character 'U' with a frequency of 37. - **Right Sub-child (42)**: Represents the character 'A' with a frequency of 42. - **Right Child (107)**: - **Left Sub-child (42)**: Represents the character 'L' with a frequency of 42. - **Right Sub-child (65)**: Breaks down further. 4. **Third Level**: - From node 65: - **Left Child (32)**: Represents the character 'C' with a frequency of 32. - **Right Child (33)**: - **Left Sub-child (9)**: - **Left Sub-child (2)**: Represents the character 'Z' with a frequency of 2. - **Right Sub-child (7)**: Represents the character 'K' with a frequency of 7. - **Right Sub-child (24)**: Represents the character 'M' with a frequency of 24. ### Edge Values: Each edge connecting the nodes is labeled with a binary digit (0 or 1), which is used in encoding the characters: - A left branch signifies a `0`. - A right branch signifies a `1`. This binary pattern assists in creating a prefix code for efficient data compression, where no code is a prefix of another, ensuring there is only one way to decode it. Understanding Huffman trees is a crucial aspect of learning about data structures and algorithms involved in data compression tasks.
Transcribed Image Text:# The Correct Huffman Tree This diagram illustrates a Huffman tree, which is a binary tree used for data compression. Each node in the tree represents a cumulative frequency, while the leaves encase characters and their respective frequencies. ### Explanation of the Tree: 1. **Root Node (306)**: The topmost node of the tree holds the total frequency of all characters combined, which is 306. 2. **First Level**: - **Left Child (120)**: This node branches out directly from the root and represents a frequency of 120 for character 'E'. - **Right Child (186)**: This node leads to further branches with a cumulative frequency of 186. 3. **Second Level**: - From node 186: - **Left Child (79)**: - **Left Sub-child (37)**: Represents the character 'U' with a frequency of 37. - **Right Sub-child (42)**: Represents the character 'A' with a frequency of 42. - **Right Child (107)**: - **Left Sub-child (42)**: Represents the character 'L' with a frequency of 42. - **Right Sub-child (65)**: Breaks down further. 4. **Third Level**: - From node 65: - **Left Child (32)**: Represents the character 'C' with a frequency of 32. - **Right Child (33)**: - **Left Sub-child (9)**: - **Left Sub-child (2)**: Represents the character 'Z' with a frequency of 2. - **Right Sub-child (7)**: Represents the character 'K' with a frequency of 7. - **Right Sub-child (24)**: Represents the character 'M' with a frequency of 24. ### Edge Values: Each edge connecting the nodes is labeled with a binary digit (0 or 1), which is used in encoding the characters: - A left branch signifies a `0`. - A right branch signifies a `1`. This binary pattern assists in creating a prefix code for efficient data compression, where no code is a prefix of another, ensuring there is only one way to decode it. Understanding Huffman trees is a crucial aspect of learning about data structures and algorithms involved in data compression tasks.
**Huffman Coding and Tree Construction** **Task:** Given the following letter frequencies, construct a Huffman Tree and derive the corresponding Huffman Codes. **Letter Frequencies:** - c: 32 - d: 42 - e: 120 - k: 7 - l: 42 - m: 24 - u: 37 - z: 2 **Steps to Construct the Huffman Tree:** 1. **Initial Nodes:** - Start with each letter as a node with its frequency. 2. **Combine Nodes with Lowest Frequencies:** - Combine the two nodes with the smallest frequencies to create a new node. The frequency of the new node is the sum of the two nodes combined. - This process is repeated, building upward until all nodes are combined into a single tree. 3. **Assign Codes:** - Starting from the root, traverse the tree to assign binary codes. Assign '0' for the left branch and '1' for the right branch. - Each leaf node (representing a letter) will have a unique binary code. **Example of Steps for This Problem:** 1. Begin with nodes for each letter: - z:2, k:7, m:24, c:32, u:37, d:42, l:42, e:120 2. Combine the nodes with the smallest frequencies: - z (2) and k (7) → New node with frequency 9 3. Proceed similarly until the tree is fully constructed. 4. The final tree will provide a Huffman Code for each letter, minimizing the total number of bits needed for representation based on the frequency of each letter. By constructing the Huffman Tree using this method, you achieve optimal prefix codes suitable for data compression applications.
Transcribed Image Text:**Huffman Coding and Tree Construction** **Task:** Given the following letter frequencies, construct a Huffman Tree and derive the corresponding Huffman Codes. **Letter Frequencies:** - c: 32 - d: 42 - e: 120 - k: 7 - l: 42 - m: 24 - u: 37 - z: 2 **Steps to Construct the Huffman Tree:** 1. **Initial Nodes:** - Start with each letter as a node with its frequency. 2. **Combine Nodes with Lowest Frequencies:** - Combine the two nodes with the smallest frequencies to create a new node. The frequency of the new node is the sum of the two nodes combined. - This process is repeated, building upward until all nodes are combined into a single tree. 3. **Assign Codes:** - Starting from the root, traverse the tree to assign binary codes. Assign '0' for the left branch and '1' for the right branch. - Each leaf node (representing a letter) will have a unique binary code. **Example of Steps for This Problem:** 1. Begin with nodes for each letter: - z:2, k:7, m:24, c:32, u:37, d:42, l:42, e:120 2. Combine the nodes with the smallest frequencies: - z (2) and k (7) → New node with frequency 9 3. Proceed similarly until the tree is fully constructed. 4. The final tree will provide a Huffman Code for each letter, minimizing the total number of bits needed for representation based on the frequency of each letter. By constructing the Huffman Tree using this method, you achieve optimal prefix codes suitable for data compression applications.
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