A Practical Introduction to Data Structures and Algorithm Analysis

190 Chap. 5 Binary TreesLetter C D E K L M U ZFrequency 32 42 120 7 42 24 37 2Figure 5.24 The relative frequencies for eight selected letters.the weighted path length of a leaf to be its weight times its depth. The binary treewith minimum external path weight is the one with the minimum sum of weightedpath lengths for the given set of leaves. A letter with high weight should have lowdepth, so that it will count the least against the total path length. As a result, anotherletter might be pushed deeper in the tree if it has less weight.The process of building the Huffman tree for n letters is quite simple. First,create a collection of n initial Huffman trees, each of which is a single leaf nodecontaining one of the letters. Put the n partial trees onto a min-heap (a priorityqueue) organized by weight (frequency). Next, remove the first two trees (the oneswith lowest weight) from the heap. Join these two trees together to create a newtree whose root has the two trees as children, and whose weight is the sum ofthe weights of the two trees. Put this new tree back on the heap. This process isrepeated until all of the partial Huffman trees have been combined into one.Example 5.8 Figure 5.25 illustrates part of the Huffman tree constructionprocess for the eight letters of Figure 5.24. Ranking D and L arbitrarily byalphabetical order, the letters are ordered by frequency asLetter Z K M C U D L EFrequency 2 7 24 32 37 42 42 120Because the first two letters on the list are Z and K, they are selected tobe the first trees joined together. They become the children of a root nodewith weight 9. Thus, a tree whose root has weight 9 is placed back on thelist, where it takes up the first position. The next step is to take values 9and 24 off the heap (corresponding to the partial tree with two leaf nodesbuilt in the last step, and the partial tree storing the letter M, respectively)and join them together. The resulting root node has weight 33, and so thistree is placed back into the heap. Its priority will be between the trees withvalues 32 (for letter C) and 37 (for letter U). This process continues until atree whose root has weight 306 is built. This tree is shown in Figure 5.26.Figure 5.27 shows an implementation for Huffman tree nodes. This implementationis similar to the VarBinNode implementation of Figure 5.10. There is anabstract base class, named HuffNode, and two subclasses, named LeafNode