Disordered Binary Search Trees
March 10, 2017
In a recent exercise we wrote a program to determine if a binary tree was balanced. Today’s exercise is similar:
Write a program to determine if a binary tree satisfies the binary search tree property that all left children are less than their parent and all right children are greater than their parent.
Your task is to write a program to determine if a binary tree is a binary search tree. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.
Programming Praxis 
@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 TrueUnfortunately 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
ANother try to print the tree
An in-order traversal of a valid binary search tree will visit the nodes in sorted order. An in-order traversal of an invalid binary search tree will visit the nodes in unsorted order.
Here’s a solution in Java that validates a binary search tree by considering whether nodes visited in-order are sorted. Morris traversal is used for O(1) space, O(n) runtime.
The example checks a valid and invalid tree.
public class BinarySearchTree<T extends Comparable<T>> { class Node { T value; Node left; Node right; Node(T value) { this.value = value; } } Node root; private Node insert(Node node, Node root) { if (root == null) return node; int compare = node.value.compareTo(root.value); if (compare <= 0) { root.left = insert(node, root.left); } else { root.right = insert(node, root.right); } return root; } Node insert(T value) { Node node = new Node(value); root = insert(node, root); return node; } class Validator { T lastValue; boolean validate(T value) { boolean valid = true; if (lastValue != null) { if (value.compareTo(lastValue) < 0) { valid = false; } } lastValue = value; return valid; } } boolean valid() { Validator validator = new Validator(); Node node = root; while (node != null) { if (node.left == null) { if (!validator.validate(node.value)) { return false; } node = node.right; } else { Node predecessor = node.left; while (predecessor.right != null && node != predecessor.right) { predecessor = predecessor.right; } if (node == predecessor.right) { if (!validator.validate(node.value)) { return false; } predecessor.right = null; node = node.right; } else { predecessor.right = node; node = node.left; } } } return true; } public String toString(String prefix, Node node, boolean right) { if (node == null) return ""; StringBuffer sb = new StringBuffer(); sb.append(toString(prefix + (right ? " " : "│ "), node.right, true)); sb.append(prefix + (right ? "┌── " : "└── ") + node.value + "\n"); sb.append(toString(prefix + (right ? "│ " : " "), node.left, false)); return sb.toString(); } public String toString() { return toString("", root, false); } public static void main(String[] args) { BinarySearchTree<Integer> bst = new BinarySearchTree<>(); int[] values = {43,2,5,42,12,672,3,34}; BinarySearchTree.Node[] nodes = new BinarySearchTree.Node[values.length]; for (int i = 0; i < values.length; ++i) { nodes[i] = bst.insert(values[i]); } System.out.println("Original"); System.out.print(bst); System.out.format("Valid: %s\n", bst.valid()); // Modify tree to invalidate bst.root.left.value = bst.root.value + 1; System.out.println(); System.out.println("Modified"); System.out.print(bst); System.out.format("Valid: %s\n", bst.valid()); } }Output:
Original │ ┌── 672 └── 43 │ ┌── 42 │ │ │ ┌── 34 │ │ └── 12 │ ┌── 5 │ │ └── 3 └── 2 Valid: true Modified │ ┌── 672 └── 43 │ ┌── 42 │ │ │ ┌── 34 │ │ └── 12 │ ┌── 5 │ │ └── 3 └── 44 Valid: false