1. BINARY SEARCH TREE

2. Binary Trees A binary tree is a tree in which no node can have more than two children. In this case we can keep direct links to the children: struct TreeNode { Object element; TreeNode *left_child; TreeNode *right_child; };

3. Binary Trees: Recursive Definition ROOT OF TREE T T1 T2 SUBTREES *left_child *right_child

4. Binary Search Trees Assume that each node in the tree stores an item (e.g., a word from a dictionary). These items can be ordered in some consistent manner. A binary search tree is a binary tree where, for every node X in the tree, the values of the items in its left subtree are smaller than the item in X, and the values of the items in its right subtree are larger than the item in X.

5. Binary Search Tree ROOT OF TREE T T1 T2 SUBTREES *left_child *right_child X All nodes in T1 have values < X. All nodes in T2 have values > X.

6. Binary Search Trees in C++ We will use two classes: The class BinaryNode simply constructs individual nodes in the tree. The class BinarySearchTree maintains a pointer to the root of the binary search tree and includes methods for inserting and removing nodes.

7. Binary Search Trees in C++ element, *left, *right constructor method The BinaryNode Class class BinaryNode { int element; BinaryNode *left; BinaryNode *right; BinaryNode (const int & el, BinaryNode *lt, BinaryNode *rt) : element (el), left (lt), right (rt) { } friend class BinarySearchTree; }; *left*right el

8. Binary Search Trees in C++ *root find/insert/remove methods The BinarySearchTree Class class BinarySearchTree { public: explicit BinarySearchTree (const int &notFound); const int &find (const int &x) int; void insert (const int &x); void remove (const int &x); private: BinaryNode *root; const int NOT_FOUND; }; *left*right el *root

9. Binary Search Tree Methods The Find operation returns a pointer to the node in a tree T that has item X, or NULL if there is no such node. The Insert operation inserts a new node (with item X) into the tree T. The Remove operation removes the node (with item X) from the tree T.

10. The Find Operation // Returns the element at the node const int &elementAt (BinaryNode *t) const { return t == NULL ? NOT_FOUND : t  element; } // Start the search at the root node const int &find (const int & x) const { return elementAt (find (x, root)); } element *t *root

11. The Find Operation… BinaryNode *find (const int &x, BinaryNode *t) const { if ( t == NULL ) return NULL; else if ( x < t  element ) return find ( x, t  left ); else if ( t  element < x ) return find ( x, t  right ); else return t; // Match } *root 6 2 5 9 710 Suppose I try to find the node with 7 in it. First go down the right subtree, then go down the left subtree.

12. The FindMin Operation… BinaryNode *findMin (BinaryNode *t) const { if ( t == NULL ) return NULL; else if ( t  left == NULL ) return t; return findMin (t  left); } *root 6 2 5 9 710 This function returns a pointer to the node containing the smallest element in the tree. It does so by following the left side of the tree.

13. The FindMax Operation… BinaryNode *findMax (BinaryNode *t) const { if ( t == NULL ) return NULL; else if ( t  right == NULL ) return t; return findMax (t  right); } *root 6 2 5 9 710 This function returns a pointer to the node containing the largest element in the tree. It does so by following the right side of the tree.

14. The Insert Operation Insertion is conceptually simple. To insert X into a tree T, proceed down the tree as you would with a find. If X is found, do nothing. Otherwise insert X at the last spot on the path that has been traversed. *root 6 2 5 9 710 1 For example, suppose I want to insert a 1 into this tree…

15. The Insert Operation void BinarySearchTree insert (const int &x, BinaryNode *&t) const { if (t == NULL) t = new BinaryNode (x, NULL, NULL); else if (x < t  element) insert(x, t  left); else if( t  element < x) insert(x, t  right); else ; // Duplicate entry; do nothing } *root 6 2 5 9 710 t  left 1 NULL t Note the pointer t is passed using call by reference. In this case this means t  left will be changed to t.

16. The Removal Operation If the node to be removed is a leaf, it can be deleted immediately. If the node has one child, the node can be deleted after its parent adjusts a link to bypass the deleted node. *root 6 2 5 9 710 What if the 2 is deleted?

17. Removal… *root 6 2 5 9 710 *root 6 2 5 9 710 t t  right Set t = t  right

18. Removal… If the node to be removed has two children, the general strategy is to replace the data of this node with the smallest data of the right subtree. Then the node with the smallest data is now removed (this case is easy since this node cannot have two children).

19. Removal… *root 6 2 5 9 7101 3 4 Remove the 2 again… *root 6 3 5 9 7101 3 4

20. The Removal Operation void remove (const int &x, BinaryNode *&t) const { if ( t == NULL ) return; // Item not found; do nothing if ( x < t  element ) remove( x, t  left ); else if( t  element < x ) remove( x, t  right ); else if( t  left != NULL && t  right != NULL ) // Two children { t  element = findMin( t  right )  element; remove( t  element, t  right ); } else // One child { BinaryNode *oldNode = t; t = ( t  left != NULL ) ? t  left : t  right; delete oldNode; } }

21. Other Methods Please see the text for the C++ destructor and copy constructors (pages 139-140).

22. Analysis The running time of these operations is O(d), where d is the depth of the node containing the accessed item. What is the average depth of the nodes in a binary search tree? It depends on how well balanced the tree is.

23. Average Depth of Nodes 10 520 181334 Consider this very well-balanced binary search tree. What is the depth of its leaf nodes? N=7 Data Order: 10, 5, 1, 8, 20, 13, 34

24. A Better Analysis The analysis on the previous slide was for a particularly well-balanced binary search tree. However, not all binary search trees will be this well balanced. In particular, binary search trees are created via insertions of data. Depending on the order of the data, various trees will emerge.

25. Effect of Data Order 4 3 2 1 Obtained if data is 4, 3, 2 1 1 2 3 4 Obtained if data is 1, 2, 3, 4 Note in these cases the average depth of nodes is about N/2, not log(N)!

26. Depth of Nodes In the best case the depth will be about O(log N). In the worst case, if the data are already ordered, the depth will be about O(N).

27. Effects of Data Order… So, if the input data are randomly ordered, what is the average depth of the nodes? The analysis is beyond the scope of this course, but it can be shown that the average depth is O(log N), which is a very nice result.

28. Summary In this lecture we showed that, for an average binary search tree, the average depth of the nodes is O(log N). This is quite amazing, indicating that the bad situations, which are O(N), don’t occur very often. However, for those who are still concerned about the very bad situations, we can try to “balance” the trees. This is the subject for the next lecture.