**What is the Length of the Weighted Path of the Huffman Tree?**
**1. Tree Path Length**
The path length of a tree refers to the total number of edges from the root node to each leaf node. In a binary tree with the same number of nodes, a complete binary tree has the shortest path length. This concept is important in understanding how efficiently data is stored or accessed in tree structures.
**2. Weighted Path Length of the Tree (WPL)**
In some applications, each node in a tree can be assigned a weight, which represents a specific value or importance. The weighted path length for a node is calculated as the product of its path length from the root and its weight. The overall weighted path length of the tree (WPL) is the sum of the weighted path lengths of all the leaf nodes. It is commonly expressed as:
$$ WPL = \sum_{i=1}^{n} w_i \times l_i $$
Where:
- $ n $ is the number of leaf nodes.
- $ w_i $ is the weight of the $ i $-th leaf node.
- $ l_i $ is the path length from the root to the $ i $-th leaf node.
The WPL is also referred to as the cost of the tree. A lower WPL means the tree is more efficient in terms of storage or retrieval.
**3. Optimal Binary Tree or Huffman Tree**
An optimal binary tree, also known as a Huffman tree, is a binary tree constructed from $ n $ leaf nodes with given weights $ w_1, w_2, ..., w_n $. Among all possible binary trees, the one with the smallest WPL is considered the optimal one.
For example, suppose we have four leaf nodes with weights 7, 5, 2, and 4. Three different binary trees can be constructed, and their WPLs are as follows:
- (a) WPL = 7×2 + 5×2 + 2×2 + 4×2 = 36
- (b) WPL = 7×3 + 5×3 + 2×1 + 4×2 = 46
- (c) WPL = 7×1 + 5×2 + 2×3 + 4×3 = 35
Tree (c) has the smallest WPL and is therefore the Huffman tree.
**Notes:**
1. When all leaf weights are equal, a complete binary tree is always the optimal one. However, this isn’t always the case when weights differ.
2. In an optimal binary tree, nodes with larger weights tend to be closer to the root.
3. There may be multiple valid Huffman trees, but they will all have the same minimum WPL.
**How to Calculate the Weighted Path Length of a Huffman Tree**
**[Problem Description]**
You are given two lines of input. The first line contains a positive integer indicating the number of leaf nodes. The second line contains the weights of these leaf nodes, separated by spaces. Your task is to construct a Huffman tree and calculate the weighted path length (WPL) of the tree.
**[Input Format]**
The first line contains an integer $ n $, representing the number of leaf nodes. The second line contains $ n $ positive integers, representing the weights of the leaf nodes. The number of nodes does not exceed 1000.
**[Output Format]**
Output the corresponding weighted path length of the Huffman tree.
**[Sample Input]**
5
4 5 6 7 8
**[Sample Output]**
69
**About the Huffman Tree**
**1. Path Length**
A path in a tree is formed by traversing from one node to another through branches. The number of branches in a path is called the path length. For example, in a binary tree, the path from the root to a leaf node may consist of several steps, and the number of those steps defines the path length.
**2. Tree Path Length**
The path length of the entire tree is the sum of the path lengths from the root to each node. This helps determine the efficiency of traversal in the tree structure.
Huffman trees are widely used in data compression algorithms, such as Huffman coding, where the goal is to minimize the average code length based on the frequency of symbols. By constructing a tree that minimizes the WPL, we ensure that the most frequent elements have shorter paths, leading to more efficient encoding.
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