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SICP exercise 2.65

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Use the results of exercises 2.63 and 2.64 to give θ(n) implementations of union-set and intersection-set for sets implemented as (balanced) binary trees.


(define (entry tree) (car tree))
(define (left-branch tree) (cadr tree))
(define (right-branch tree) (caddr tree))
(define (make-tree entry left right)
  (list entry left right))
(define (tree->list tree)
  (define (copy-to-list tree result-list)
    (if (null? tree)
        (copy-to-list (left-branch tree)
                      (cons (entry tree)
                            (copy-to-list (right-branch tree)
  (copy-to-list tree '()))
(define (list->tree elements)
  (car (partial-tree elements (length elements))))
(define (partial-tree elts n)
  (if (= n 0)
      (cons '() elts)
      (let ((left-size (quotient (- n 1) 2)))
        (let ((left-result (partial-tree elts left-size)))
          (let ((left-tree (car left-result))
                (non-left-elts (cdr left-result))
                (right-size (- n (+ left-size 1))))
            (let ((this-entry (car non-left-elts))
                  (right-result (partial-tree (cdr non-left-elts)
              (let ((right-tree (car right-result))
                    (remaining-elts (cdr right-result)))
                (cons (make-tree this-entry left-tree right-tree)
(define (union-set set1 set2)
  (define (union-set-list list1 list2)
    (cond ((null? list1) list2)
          ((null? list2) list1)
          (else (let ((x1 (car list1))
                      (x2 (car list2)))
                  (cond ((= x1 x2)
                         (cons x1 (union-set-list (cdr list1) (cdr list2))))
                        ((< x1 x2)
                         (cons x1 (union-set-list (cdr list1) list2)))
                         (cons x2 (union-set-list list1 (cdr list2)))))))))
  (list->tree (union-set-list (tree->list set1)
                              (tree->list set2))))
(define (intersection-set set1 set2)
  (define (intersection-set-list list1 list2) 
    (if (or (null? list1) (null? list2)) 
        (let ((x1 (car list1)) (x2 (car list2))) 
          (cond ((= x1 x2) 
                 (cons x1 
                       (intersection-set-list (cdr list1) 
                                         (cdr list2)))) 
                ((< x1 x2) 
                 (intersection-set-list (cdr list1) list2)) 
                ((< x2 x1) 
                 (intersection-set-list list1 (cdr list2))))))) 
  (list->tree (intersection-set-list (tree->list set1)
                                     (tree->list set2))))


(define s1 (list->tree '(1 2 3 5 6 8 10 11)))
(define s2 (list->tree '(2 4 6 8 11 13)))
(tree->list (union-set s1 s2))


(1 2 3 4 5 6 8 10 11 13)
(tree->list (intersection-set s1 s2))


(2 6 8 11)

My solutions transform the tree to a sorted list, perform the algorithms developed in the text and in exercise 2.62, and then transform the sorted list back into a tree.

Each procedure used by union-set or intersection-set to perform the operations is θ(n). Except for the tree->list transformation, which is called twice per application of union-set or intersection-set, each procedure is called once per application, so those implementations are also θ(n).

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