Disordered Binary Search Trees

March 10, 2017

As in the past, a binary tree is a list of three nodes with the value in the car, the left child in the cadr, and the right child in the caddr.

A null tree is a binary search tree. A non-null tree is a binary search tree if its left child is a binary search tree, its right child is a binary search tree, its node value is greater the head of its left child, and its node value is less than the head of its right child:

(define (ordered? lt? tree)
  (or (null? tree)
      (and (ordered? lt? (cadr tree))
           (ordered? lt? (caddr tree))
           (or (null? (cadr tree))
               (lt? (caadr tree) (car tree)))
           (or (null? (caddr tree))
               (lt? (car tree) (caaddr tree))))))

Here are two examples:

> (ordered? < '(4 (0 () (1 () (2 () (3 () ())))) (5 () (6 () ())))) #t > (ordered? < '(6 (0 () (1 () (2 () (3 () ())))) (5 () (4 () ()))))
#f

You can run the program at http://ideone.com/i7MGDU.

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3 Responses to “Disordered Binary Search Trees”

  1. Paul said

    @programmingpraxis: If I understand your definition of a BST given in the solution, then the following tree would be a BST:

    ├── 10 (root)
    │ ├── 40
    │ └── 5
    │ ├── 20
    │ └── 1

    The tree at the left (with root value 5 and children 1 and 20) is a BST, but the total tree is not, as in the left tree the value 20 is higher than 10.

    Below an iterative solution in Python. Recursive is not a good idea for this in Python, because it would not work for large trees.

    def is_bst(root):
        Q = [(root, -inf, inf)]
        while Q:
            node, lo, hi = Q.pop()
            if not node:
                continue
            if not lo < node.value < hi:
                return False
            Q += [(node.lft, lo, node.value), (node.rgt, node.value, hi)]
        return True
    
  2. Paul said

    Unfortunately the tree is not printing well (it does in “preview”). The tree is:
    root: 10 with children 5 and 40
    left child of root: 5 with children 1 and 20

  3. Paul said

    ANother try to print the tree

    ├── 10
    │   ├── 40
    │   └── 5
    │       ├── 20
    │       └── 1
    

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