A Practical Introduction to Data Structures and Algorithm Analysis

162 Chap. 5 Binary Trees205040 7520 to 40Figure 5.6 To be a binary search tree, the left child of the node with value 40must have a value between 20 and 40.fall within a given range of values. Fortunately, it requires only a simple localcalculation to determine which child(ren) to visit.A more difficult situation is illustrated by the following problem. Given anarbitrary binary tree we wish to determine if, for every node A, are all nodes in A’sleft subtree less than the value of A, and are all nodes in A’s right subtree greaterthan the value of A? (This happens to be the definition for a binary search tree,described in Section 5.4.) Unfortunately, to make this decision we need to knowsome context that is not available just by looking at the node’s parent or children.As shown by Figure 5.6, it is not enough to verify that A’s left child has a valueless than that of A, and that A’s right child has a greater value. Nor is it enough toverify that A has a value consistent with that of its parent. In fact, we need to knowinformation about what range of values is legal for a given node. That informationmight come from any ancestor for the node. Thus, relevant range information mustbe passed down the tree. We can implement this function as follows.boolean checkBST(BSTNode root,Integer low, Integer high) {if (root == null) return true; // Empty subtreeInteger rootkey = root.key();if ((rootkey < low) || (rootkey > high))return false; // Out of rangeif (!checkBST(root.left(), low, rootkey))return false; // Left side failedreturn checkBST(root.right(), rootkey, high);}5.3 Binary Tree Node ImplementationsIn this section we will examine ways to implement binary tree nodes. We begin withsome options for pointer-based binary tree node implementations. Then comes a