**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 the tree. In a binary tree with the same number of nodes, a complete binary tree has the shortest possible path length. This makes it more efficient in terms of traversal and storage.
**2. Weighted Path Length of the Tree (WPL)**
In some applications, nodes in a tree can have associated weights—real numbers that represent specific values or priorities. The weighted path length of a node is calculated as the product of the path length from the node to the root and the weight of that node.
The **Weighted Path Length of the Tree (WPL)** is defined as the sum of the weighted path lengths of all the **leaf nodes** in the tree. It is usually 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
This value is also referred to as the **cost of the tree**, and minimizing it is the goal when constructing an optimal binary tree.
**3. Optimal Binary Tree or Huffman Tree**
Among all binary trees that can be formed using $ n $ leaf nodes with given weights $ w_1, w_2, ..., w_n $, the one with the **smallest weighted path length** is known as the **optimal binary tree** or **Huffman tree**.
For example, consider 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 is not always the case when weights differ.
2. In an optimal binary tree, **heavier nodes tend to be closer to the root**, which helps reduce the overall WPL.
3. There may be multiple valid Huffman trees, but they will all have the **same minimal 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, and the second line contains a list of positive integers representing the weights of these nodes. Your task is to construct a Huffman tree and compute the weighted path length (WPL) of the tree.
**Input Format**
- The first line contains an integer $ n $ (number of leaf nodes).
- The second line contains $ n $ space-separated integers representing the weights of the leaf nodes.
**Output Format**
- Output the computed WPL value.
**Sample Input**
```
5
4 5 6 7 8
```
**Sample Output**
```
69
```
**About the Huffman Tree**
**Path Length**
A path in a tree is a sequence of edges connecting two nodes. The **path length** between two nodes is the number of edges in that path. For example, the path length from the root to a leaf node is the number of steps needed to reach that node from the top of the tree.
**Tree Path Length**
The path length of a tree is the sum of the path lengths from the root to each individual node. In a full binary tree, this value is minimized when the tree is balanced.
Understanding how to calculate the WPL is essential for applications like data compression, where efficiency and minimization of cost are critical. By building a Huffman tree, we ensure that the most frequent (or heavily weighted) elements are placed closer to the root, reducing the average path length and improving performance.
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