Data Structures and Algorithm Analysis - Computer Science at ...

Sec. 5.4 Binary Search Trees 163• Right sibling(r) = r + 1 if r is odd and r + 1 < n.5.4 Binary Search TreesSection 4.4 presented the dictionary ADT, along with dictionary implementationsbased on sorted and unsorted lists. When implementing the dictionary with anunsorted list, inserting a new record into the dictionary can be performed quickly byputting it at the end of the list. However, searching an unsorted list for a particularrecord requires Θ(n) time in the average case. For a large database, this is probablymuch too slow. Alternatively, the records can be stored in a sorted list. If the listis implemented using a linked list, then no speedup to the search operation willresult from storing the records in sorted order. On the other hand, if we use a sortedarray-based list to implement the dictionary, then binary search can be used to finda record in only Θ(log n) time. However, insertion will now require Θ(n) time onaverage because, once the proper location for the new record in the sorted list hasbeen found, many records might be shifted to make room for the new record.Is there some way to organize a collection of records so that inserting recordsand searching for records can both be done quickly? This section presents thebinary search tree (BST), which allows an improved solution to this problem.A BST is a binary tree that conforms to the following condition, known asthe Binary Search Tree Property: All nodes stored in the left subtree of a nodewhose key value is K have key values less than K. All nodes stored in the rightsubtree of a node whose key value is K have key values greater than or equal to K.Figure 5.13 shows two BSTs for a collection of values. One consequence of theBinary Search Tree Property is that if the BST nodes are printed using an inordertraversal (see Section 5.2), the resulting enumeration will be in sorted order fromlowest to highest.Figure 5.14 shows a class declaration for the BST that implements the dictionaryADT. The public member functions include those required by the dictionaryADT, along with a constructor and destructor. Recall from the discussion in Section4.4 that there are various ways to deal with keys and comparing records (threeapproaches being key/value pairs, a special comparison method such as using theComparator class, and passing in a comparator function). Our BST implementationwill handle comparison by explicitly storing a key separate from the data valueat each node of the tree.To find a record with key value K in a BST, begin at the root. If the root storesa record with key value K, then the search is over. If not, then we must searchdeeper in the tree. What makes the BST efficient during search is that we needsearch only one of the node’s two subtrees. If K is less than the root node’s keyvalue, we search only the left subtree. If K is greater than the root node’s keyvalue, we search only the right subtree. This process continues until a record with