Optimal binary tree (Huffman tree)

Pre-Knowledge:

　　path : A path from one node in the tree to another, and the number of branches on the path becomes the length of the route;

the path length of the tree : the sum of the length of the path from the root to each leaf;

　　The belt-weighted path length of a node: the product of the path length from the node to the root of the tree and the value of the node.

　　the length of the weighted path of the tree : The sum of the weighted path length of all leaf nodes;

Any two binary trees with the same structure, because their leaf node weights may be different, so their path length will be different; Huffman tree refers to the two-tree with the smallest length of the weighted path;

How to construct Huffman tree:

Suppose there is a set of weights {5,29,7,8,14,23,3,11}, we will use this set of weights to demonstrate the process of constructing Huffman tree.

The first step: to construct a forest with 8 trees with the weights of the 8 weights of the seat root node;

　　

　　The second step: choose two root of the smallest weight of the tree 3, 5 seats left and right sub-tree to form a new tree, the new root node of the weight of 3+5, and the two trees removed from the forest, and the new tree added;

　　Step three: Repeat the second step until there is only one tree in the forest, select 7,8 here;

　　Fourth Step:

Fifth Step:

Sixth step:

　　Seventh Step:

Huffman Code:

In Huffman tree, there are only nodes with leaf node and degree 2. Number of leaf nodes = +1 of nodes with degrees 2.

Optimal binary tree (Huffman tree)