A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 5.6 Huffman Coding Trees 193/** Binary tree node implementation with just an elementfield (no key) and pointers to children */class HuffNode implements BinNode {private E element;// Element for this nodeprivate HuffNode left; // Pointer to left childprivate HuffNode right; // Pointer to right child}/** Constructors */public HuffNode() {left = right = null; }public HuffNode(E val){ left = right = null; element = val; }public HuffNode(E val, HuffNode l, HuffNode r){ left = l; right = r; element = val; }/** Return and set the element value */public E element() { return element; }public void setElement(E v) { element = v; }/** Return and set the left child */public HuffNode left() { return left; }public HuffNode setLeft(HuffNode p){ return left = p; }/** Return and set the right child */public HuffNode right() { return right; }public HuffNode setRight(HuffNode p){ return right = p; }/** Return true if this is a leaf node */public boolean isLeaf(){ return (left == null) && (right == null); }Figure 5.27 Implementation for Huffman tree nodes. Internal nodes and leafnodes are represented by separate classes, each derived from an abstract base class.have selected node V in place of node X as the child of node U. However, this isimpossible because l 1 and l 2 are the letters with least frequency.✷Theorem 5.3 Function buildHuff builds the Huffman tree with the minimumexternal path weight for the given set of letters.Proof: The proof is by induction on n, the number of letters.• Base Case: For n = 2, the Huffman tree must have the minimum externalpath weight because there are only two possible trees, each with identicalweighted path lengths for the two leaves.