Encode-symbol for Huffman tree From the text:


  Exercise 2.68.  The encode procedure
  takes as arguments a message and a
  tree and produces the list of bits
  that gives the encoded message.


(define (encode message tree)
  (if (null? message)
      '()
      (append (encode-symbol (car message) tree)
              (encode (cdr message) tree))))



  Encode-symbol is a procedure, which
  you must write, that returns the list
  of bits that encodes a given symbol
  according to a given tree. You should
  design encode-symbol so that it
  signals an error if the symbol is not
  in the tree at all. Test your
  procedure by encoding the result you
  obtained in exercise 2.67 with the
  sample tree and seeing whether it is
  the same as the original sample
  message.


The textbook provides the following definitions:

(define (make-leaf symbol weight)
  (list 'leaf symbol weight))
(define (leaf? object)
  (eq? (car object) 'leaf))
(define (symbol-leaf x) (cadr x))
(define (weight-leaf x) (caddr x))

(define (make-code-tree left right)
  (list left
        right
        (append (symbols left) (symbols right))
        (+ (weight left) (weight right))))

(define (left-branch tree) (car tree))

(define (right-branch tree) (cadr tree))
(define (symbols tree)
  (if (leaf? tree)
      (list (symbol-leaf tree))
      (caddr tree)))

(define (weight tree)
  (if (leaf? tree)
      (weight-leaf tree)
      (cadddr tree)))

(define (decode bits tree)
  (define (decode-1 bits current-branch)
    (if (null? bits)
        '()
        (let ((next-branch
               (choose-branch (car bits) current-branch)))
          (if (leaf? next-branch)
              (cons (symbol-leaf next-branch)
                    (decode-1 (cdr bits) tree))
              (decode-1 (cdr bits) next-branch)))))
  (decode-1 bits tree))
(define (choose-branch bit branch)
  (cond ((= bit 0) (left-branch branch))
        ((= bit 1) (right-branch branch))
        (else (error "bad bit -- CHOOSE-BRANCH" bit))))
(define (adjoin-set x set)
  (cond ((null? set) (list x))
        ((&lt; (weight x) (weight (car set))) (cons x set))
        (else (cons (car set)
                    (adjoin-set x (cdr set))))))

(define (make-leaf-set pairs)
  (if (null? pairs)
      '()
      (let ((pair (car pairs)))
        (adjoin-set (make-leaf (car pair)    ; symbol
                               (cadr pair))  ; frequency
                    (make-leaf-set (cdr pairs))))))

(define sample-tree
  (make-code-tree (make-leaf 'A 4)
                  (make-code-tree
                   (make-leaf 'B 2)
                   (make-code-tree (make-leaf 'D 1)
                                   (make-leaf 'C 1)))))

(define sample-message '(0 1 1 0 0 1 0 1 0 1 1 1 0))

(define (encode message tree)
  (if (null? message)
      '()
      (append (encode-symbol (car message) tree)
              (encode (cdr message) tree))))


And I wrote the following answer:

(define (member? x set)
  (not (equal? (member x set) false)))

(define (encode-branch symbol tree)
  (let ((left (left-branch tree))
        (right (right-branch tree)))
    (cond ((member? symbol (symbols left)) (list 0 left))
          ((member? symbol (symbols right)) (list 1 right))
          (else (error "Symbol is not a member of either left or right branches of the tree - can't encode" symbol tree)))))

(define (encode-symbol symbol tree)
  (if (leaf? tree) '()
      (let ((new-branch (encode-branch symbol tree)))
        (cons (car new-branch) (encode-symbol symbol (cadr new-branch))))))


How can I improve my solution?
Wouldn't it be easier to unpack the tree into a dictionary mapping each symbol to its corresponding bit string?  Then you could simply look up each symbol in the input to generate the corresponding output bits.

EDIT:
As suggested by syb0rg, here is an implementation (C#, I'm afraid -- my Lisp is far too rusty -- although it's almost pure).  The part pertaining to my suggestion above lives in the HuffmanCodes function at the end.

void Main()
{
    var corpus = @"Lorem ipsum dolor sit amet, consectetuer adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. Ut wisi enim ad minim veniam, quis nostrud exerci tation ullamcorper suscipit lobortis nisl ut aliquip ex ea commodo consequat.";
    var hcs = HuffmanCodes(corpus);
    Console.WriteLine(hcs);
}

Dictionary&lt;char, int&gt; Histogram(string corpus) {
    var hg = new Dictionary&lt;char, int&gt;();
    foreach (var x in corpus) {
        int f;
        hg.TryGetValue(x, out f);
        hg[x] = f + 1;
    }
    return hg;
}

class HuffTree {
    internal char? Sym; // Non-null iff this is a leaf node with no children.
    internal int Freq;
    internal HuffTree L;
    internal HuffTree R;
}

// Oh, for a priority queue.  This is *really* inefficient!
HuffTree HuffmanTree(string corpus) {
    var hg = Histogram(corpus);
    var hts = hg.Keys.Select(x =&gt; new HuffTree { Sym = x, Freq = hg[x] }).OrderBy(t =&gt; t.Freq).ToList();
    while (2 &lt;= hts.Count) {
        var leasts = hts.Take(2).ToList();
        var l = leasts[0];
        var r = leasts[1];
        var newHt = new HuffTree { Freq = l.Freq + r.Freq, L = l, R = r };
        hts = hts.Skip(2).Concat(new HuffTree[] { newHt }).OrderBy(t =&gt; t.Freq).ToList();
    }
    return hts.First();
}

Dictionary&lt;char, string&gt; HuffmanCodes(string corpus) {
    var codes = new Dictionary&lt;char, string&gt;();
    Action&lt;HuffTree, string&gt; a = null; // Sweet recursion.
    a = (ht, code) =&gt; {
        if (ht.Sym != null) {
            codes[(char)ht.Sym] = code;
        } else {
            a(ht.L, code + "0");
            a(ht.R, code + "1");
        }
    };
    a(HuffmanTree(corpus), "");
    return codes;
}