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After completion of the third merge, the two trees of lowest weight are the single-node trees representing i and sp Figure 1310 shows how these trees are merged into the new tree with root T4 The fifth step is to merge the trees with roots e and T 3 because these trees have the two smallest weights, giving the result shown in Figure 131 1 Finally, an optimal tree, shown previously in Figure 134, is obtained by merging the two remaining trees Figure 1312 shows the optimal tree, with root T6

For some sets of regression data a linear regression model may not be appropriate. The techniques mentioned in 2 may fail to produce a satisfactory model. Or theory about the process that generated the data may suggest a different type of model. Such data require more general models. A very general model is Yi = J.Li + e i , where J.Li is the expectation of the random variable Yi and where e i is defined to be the difference (or "error")

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*" r('17'+~ p,+p R ) = r( '17'+ ~ p,+P L ) *" r( '17'- ~ p,-P L ) =

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We now provide an implementation of the Huffman coding algorithm, without attempting to perform any significant optimizations; we simply want a working program that illustrates the basic algorithmic issues After discussing the implementation we comment on possible enhancements Although significant error checking needs to be added to the program, we have not done so because we did not want to obscure the basic ideas Figure 1313 illustrates some of the header files to be used For simplicity we use maps and maintain a priority queue of (pointers to) tree nodes (recall that we are to select two trees of lowest weight) Thus we need <queue> and <functional> -and, as it turns out, Wrapperh (because we need to wrap the pointer variables to make the comparison function meaningful) We also use <algorithm> because, in one of our routines, we use the reverse method In addition to the library classes, our program consists of several additional classes Because we need to perform bit-at-a-time 110, we write wrapper classes representing bit-input and bit-output streams We write other classes to maintain character counts and create and return information about a Huffman coding tree Finally, we write a class that contains the (static) compression and uncompression functions To summarize, the classes that we write are:

Wraps an istream and provides bit-at-a-time input Wraps an ostream and provides bit-at-a-time output Maintains character counts Manipulates Huffman coding trees Contains compression and uncompression methods

r('17'+~ p,+P L )

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Yi - fJ-i Different kinds of models are obtained by making different assumptions about the structure of the expectations fJ-i and about the distribution of

The class interfaces for ibstream and obstream are similar and are shown in Figures 1314 and 1315, respectively Both work by wrapping a stream A reference to the stream is stored as a private data member Every eighth readBit of the ibstream (orwrite~it the obstream) causes a char of to be read (or written) on the underlying stream The char is stored in a buffer, appropriately named buffer, and buff erpos provides an indication of how much of the buffer is unused Implementation of ibstream is provided in Figure 1316 The getBit and setBit methods are used to access an individual bit in an 8-bit character;' they work by using bit operations (Appendix A23 describes the bit operators in more detail) In readBit, we check at line 26 to find out whether the bits in the buffer have already been used If so, we get 8 more bits at line 28, and reset the position indicator at line 31 Then we can call getBit at line 33

/ / ibstream class interface: Bit-input stream wrapper class //

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