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Sec. 5.6 Huffman Coding Trees 185Letter Freq Code BitsC 32 1110 4D 42 101 3E 120 0 1K 7 111101 6L 42 110 3M 24 11111 5U 37 100 3Z 2 111100 6Figure 5.31 The Huffman codes for the letters of Figure 5.24.nodes in the tree were deeper, we could reduce their weighted path length byswapping them with w 1 or w 2 . But the lemma tells us that no such deepernodes exist. Call T ′ the Huffman tree that is identical to T except that nodeV is replaced with a leaf node V ′ whose weight is w 1 + w 2 . By the inductionhypothesis, T ′ has minimum external path length. Returning the children toV ′ restores tree T, which must also have minimum external path length.Thus by mathematical induction, function buildHuff creates the Huffmantree with minimum external path length.✷5.6.2 Assigning and Using Huffman CodesOnce the Huffman tree has been constructed, it is an easy matter to assign codesto individual letters. Beginning at the root, we assign either a ‘0’ or a ‘1’ to eachedge in the tree. ‘0’ is assigned to edges connecting a node with its left child, and‘1’ to edges connecting a node with its right child. This process is illustrated byFigure 5.26. The Huffman code for a letter is simply a binary number determinedby the path from the root to the leaf corresponding to that letter. Thus, the codefor E is ‘0’ because the path from the root to the leaf node for E takes a single leftbranch. The code for K is ‘111101’ because the path to the node for K takes fourright branches, then a left, and finally one last right. Figure 5.31 lists the codes forall eight letters.Given codes for the letters, it is a simple matter to use these codes to encode atext message. We simply replace each letter in the string with its binary code. Alookup table can be used for this purpose.Example 5.9 Using the code generated by our example Huffman tree,the word “DEED” is represented by the bit string “10100101” and the word“MUCK” is represented by the bit string “111111001110111101.”Decoding the message is done by looking at the bits in the coded string fromleft to right until a letter is decoded. This can be done by using the Huffman tree in