1. Computer Science and Software Engineering University of Wisconsin - Platteville 10. Binary Search Tree Yan Shi CS/SE 2630 Lecture Notes Partially adopted from C++ Plus Data Structure textbook slides

2. Review: Binary Search in a Sorted Array Based List  Examines the element in the middle of the array. Is it the sought item? If so, stop searching. Is the middle element too small? Then start looking in second half of array. Is the middle element too large? Then begin looking in first half of the array.  Repeat the process in the half of the list that should be examined next.  Stop when item is found, or when there is nowhere else to look and item has not been found.

3. Example info[0] [1] [2] [3] [4] [5] [6] [7] [8] [9] 15 26 38 57 62 78 84 91 108 119 item = 45 first midPoint last info[0] [1] [2] [3] [4] [5] [6] [7] [8] [9] 15 26 38 57 62 78 84 91 108 119 first midPoint last LESS last = midPoint - 1 GREATERfirst = midPoint + 1

4. info[0] [1] [2] [3] [4] [5] [6] [7] [8] [9] 15 26 38 57 62 78 84 91 108 119 item = 45 first, midPoint, last info[0] [1] [2] [3] [4] [5] [6] [7] [8] [9] 15 26 38 57 62 78 84 91 108 119 first, last midPoint LESS last = midPoint - 1GREATERfirst = midPoint + 1

5. info[0] [1] [2] [3] [4] [5] [6] [7] [8] [9] 15 26 38 57 62 78 84 91 108 119 item = 45 last first first > last found = false

6. Motivations for Binary Search Tree  array-based list: —fast search: binary search O(log 2 N) —slow insertion/deletion  linear linked list: —fast insertion, deletion —slow search  Can we have fast search, addition and deletion? —Binary search tree

7. Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len What is a tree? Jake’s Pizza Shop

8. Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len ROOT NODE A Tree Has a Root Node

9. Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len LEAF NODES Leaf Nodes have No Children

10. Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len LEVEL 0 A Tree Has Levels

11. Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len LEVEL 1 Level One

12. Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len LEVEL 2 Level Two

13. Owner Jake Manager Chef Brad Carol Waitress Waiter Cook Helper Joyce Chris Max Len LEFT SUBTREE OF ROOT NODE A Subtree

14. Binary Tree  A binary tree is a structure in which: —Each node can have at most two children, and in which a unique path exists from the root to every other node. —The two children of a node are called the left child and the right child, if they exist.

15. Implementing a Binary Tree with Pointers and Dynamic Data Q V T K S A E L How many leaf nodes?descendant of Q?ancestors of Q?

16. Binary Search Tree (BST)  A special kind of binary tree in which: —Each node contains a distinct data value —The key values in the tree can be compared using “greater than” and “less than”, and —The key value of each node in the tree is less than every key value in its right sub-tree, and greater than every key value in its left sub-tree.  Similar as sorted list, the order is maintained when inserting new items into the tree

17. Example  Insert ‘J’ ‘E’ ‘F’ ‘T’ ‘A’ to an empty BST.  Insert ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ to an empty BST. ‘J’‘J’ ‘E’ ‘E’ ‘F’ ‘F’ ‘T’ ‘T’ ‘A’ ‘A’ ‘A’‘A’ ‘E’‘E’ ‘F’‘F’ ‘J’‘J’ ‘T’‘T’ The shape of a BST depends on its key values and their order of insertion

18. Exercise Insert nodes containing these values in this order: ‘D’ ‘B’ ‘L’ ‘Q’ ‘S’ ‘V’ ‘Z’ ‘J’‘J’ ‘E’ ‘E’ ‘A’‘A’ ‘H’‘H’ ‘T’‘T’ ‘M’ ‘M’ ‘K’‘K’ ‘P’‘P’

19. How many nodes do we have? ‘J’‘J’ ‘E’ ‘E’ ‘A’‘A’ ‘H’‘H’ ‘T’‘T’ ‘M’ ‘M’ ‘K’‘K’ ‘V’‘V’ ‘P’‘P’ ‘Z’‘Z’‘D’‘D’‘Q’‘Q’‘L’‘L’‘B’‘B’‘S’‘S’

20. Count Nodes in a BST  CountNodes Algorithm v1: if left(tree) is NULL AND right(tree) is NULL return 1 else return CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1 What if Left(tree) is NULL?

21. Count Nodes in a BST  CountNodes Algorithm v2: if (Left(tree) is NULL) AND (Right(tree) is NULL) return 1 else if Left(tree) is NULL return CountNodes(Right(tree)) + 1 else if Right(tree) is NULL return CountNodes(Left(tree)) + 1 else return CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1 What if the initial tree is NULL?

22. Count Nodes in a BST  CountNodes Algorithm v3: if tree is NULL return 0 else if (Left(tree) is NULL) AND (Right(tree) is NULL) return 1 else if Left(tree) is NULL return CountNodes(Right(tree)) + 1 else if Right(tree) is NULL return CountNodes(Left(tree)) + 1 else return CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1 Can we simplify it somehow?

23. Count Nodes in a BST  CountNodes Algorithm Final Version: if tree is NULL return 0 else return CountNodes(Left(tree)) + CountNodes(Right(tree)) + 1 Does it cover all possible situations?

24. Is ‘F’ in the binary search tree? ‘J’‘J’ ‘E’ ‘E’ ‘A’‘A’ ‘H’‘H’ ‘T’‘T’ ‘M’ ‘M’ ‘K’‘K’ ‘V’‘V’ ‘P’‘P’ ‘Z’‘Z’‘D’‘D’‘Q’‘Q’‘L’‘L’‘B’‘B’‘S’‘S’

25. Search an item in a BST  Search Algorithm  Similar algorithm for retrieving an item given its key Given a BST tree and an item to find: if tree is NULL found = false else if ( item < current node’s info ) Search( left(tree), item ) else if ( item > current node’s info ) Search( right(tree), item ) else found = true

26. Insert an item to a BST  Use recursion  Base case: —tree is NULL (empty tree or leaf node’s left/right) —CANNOT add  item already in the BST!  General case: —if item info, Insert item to left(tree) —else if item > tree->info, Insert item to right(tree)

27. The “tree” Parameter bool BST::Insert( TNode * tree, Type item ) { if ( tree == NULL ) { tree = new TNode( item ); return true; } else if ( item info ) return Insert( tree->left, item ); else if ( item > tree->info ) return Insert( tree->right, item ); else return false; } //Is it correct?

28. The “tree” parameter  Treat “tree” as a pointer pointing within the tree!  What happens when creating a new node? —tree become a real address instead of NULL!  Make “tree” a reference parameter!

29. The “tree” Parameter bool BST::Insert( TNode * & tree, Type item ) { if ( tree == NULL ) { tree = new TNode( item ); return true; } else if ( item info ) return Insert( tree->left, item ); else if ( item > tree->info ) return Insert( tree->right, item ); else return false; } //tree is a reference parameter!

30. Delete an item in the BST  Use recursion  Base case: —If item's key matches key in Info(tree), delete node pointed to by tree. —If tree is empty, cannot delete.  General case: —if item info, Delete item in left(tree) —else if item > tree->info, Delete item in right(tree) How to delete a node?

31. Code for Delete bool BST::Delete( TNode * & tree, Type item ) { if ( tree == NULL ) return false; if ( item info ) return Delete( tree->left, item ); else if ( item > tree->info ) return Delete( tree->right, item ); else { DeleteNode( tree ); return true; } we are changing the structure of the tree! // How to do this?

32. Deleting a Leaf Node

33. Deleting a Node with One Child

34. Deleting a Node with Two Children

35. Delete a Node in BST  Algorithm: if (Left(tree) is NULL) AND (Right(tree) is NULL) Set tree to NULL else if Left(tree) is NULL Set tree to Right(tree) else if Right(tree) is NULL Set tree to Left(tree) else Find predecessor Set Info(tree) to Info(predecessor) Delete predecessor Indirect Recursion!

36. How to find a predecessor?  Algorithm: if ( tree->left != NULL ) { tree = tree->left; while (tree->right != NULL) tree = tree->right; predecessor = tree->info; }

37. Tree Traversal  In-order —from smallest to largest  pre-order —current info first —then left —then right  post-order —left first —then right —then current info

38. In-Order Print  Algorithm:  How about pre-order and post-order print? if tree is not NULL InOrderPrint( left(tree) ) print tree->info InOrderPrint( right(tree) )

39. Big-3 for BST  Destructor: —need to delete all node —traverse the tree in which order?  Copy constructor and assignment operator: —How to copy a tree? —traverse the tree in which order? —How to implement operator=? post-order! pre-order!

40. Iterative Solutions for BST  Binary Search: Has( item )  Insert: Add( item )  Delete: Remove( item )

41. Iterative Solution for Search  Algorithm for search: Has( item ) Set nodePtr to tree while more elements to search if item < Info(nodePtr) Set nodePtr to Left(nodePtr) else if item > Info(nodePtr) Set nodePtr to Right(nodePtr) else return true return false

42. Iterative Solution for Insert 1.create a node to contain the new item 2.find the insertion place —similar as the previous search —need to keep track of parent as well 3.insert new node —insert as either the left or right child of parent

43. Example: Insert Insert 13 to the tree

48. Iterative Solution for Delete TNode * node, * parent; bool found = false; FindNode( item, found, node, parent ); if ( found == false ) return false; if ( node == root ) DeleteNode( root ); else if ( parent->left == node ) DeleteNode( parent->left ); else DeleteNode( parent->right ); return true; is it really iterative if we use the previous DeleteNode function? //why parent->left?

49. nodePtr and parentPtr Are External to the Tree

50. parentPtr is External to the Tree, but parentPtr-> left is an Actual Pointer in the Tree

51. Full and Complete BST  Full Binary Tree: A binary tree in which all of the leaves are on the same level and every non-leaf node has two children  Complete Binary Tree: A binary tree that is either full or full through the next-to-last level, with the leaves on the last level as far to the left as possible

52. Examples

53. Array Representation of BST For any tree node nodes[index] left child: nodes[index*2 + 1] right child: nodes[index*2 + 2] parent: nodes[(index – 1)/2]

54. Array Representation of Non- Complete BST  Use dummy values to fill the space