1. A Binary Search Tree Implementation Chapter 25 © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. Data Structures and Abstractions with Java, 4e Frank Carrano

2. Binary Search Tree For each node in a binary search tree  Node’s data is greater than all data in node’s left subtree  Node’s data is less than all data in node’s right subtree Every node in a binary search tree is the root of a binary search tree © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

3. Binary Search Tree FIGURE 25-1 A binary search tree of names © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

4. Interface for the Binary Search Tree LISTING 25-1 An interface for a search tree © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

5. Interface for the Binary Search Tree LISTING 25-1 An interface for a search tree © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

6. Understanding the Specifications Methods will use return values instead of exceptions to indicate whether an operation has failed  getEntry, returns same entry it is given to find  getEntry returns an object in tree and matches given entry according to the entry’s compareTo method © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

7. Understanding the Specifications FIGURE 25-2 Adding an entry that matches an entry already in a binary search tree © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

8. Understanding the Specifications FIGURE 25-2 Adding an entry that matches an entry already in a binary search tree © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

9. Duplicate Entries If any entry e has a duplicate entry d, we arbitrarily require that d occur in the right subtree of e’s node For each node in a binary search tree:  Data in a node is greater than data in node’s left subtree  Data in a node is less than or equal to data in node’s right subtree © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

10. Duplicate Entries FIGURE 25-3 A binary search tree with duplicate entries © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

11. Beginning the Class Definition LISTING 25-2 An outline of the class BinarySearchTree © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

12. Beginning the Class Definition LISTING 25-2 An outline of the class BinarySearchTree © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

13. Searching and Retrieving Recursive algorithm to search a binary search tree © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

14. Searching and Retrieving Algorithm that describes actual implementation more closely © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

15. Searching and Retrieving The method getEntry uses findEntry © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

16. Searching and Retrieving The method getEntry uses findEntry © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

17. Searching and Retrieving Method contains can simply call getEntry to see whether a given entry is in the tree: © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

18. Adding an Entry FIGURE 25-4 (a) A binary search tree; © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

19. Adding an Entry FIGURE 25-4 (b) the same tree after adding Chad © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

20. Recursive Implementation FIGURE 25-5 Recursively adding Chad to smaller subtrees of a binary search tree © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

21. Recursive Implementation FIGURE 25-5 Recursively adding Chad to smaller subtrees of a binary search tree © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

22. Recursive Implementation Recursive algorithm for adding a new entry © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

23. Recursive Implementation Recursive algorithm for adding a new entry © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

24. Recursive Implementation Handle the addition to an empty binary search tree as a special case © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

25. Recursive Implementation Private recursive method addEntry © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

26. Recursive Implementation The public method add © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

27. Iterative Implementation Iterative algorithm for adding a new entry © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

28. Iterative Implementation Iterative algorithm for adding a new entry © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

29. Iterative Implementation Iterative algorithm for adding a new entry © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

30. Iterative Implementation Iterative implementation of the method addEntry © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

31. Iterative Implementation Iterative implementation of the method addEntry © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

32. Removing an Entry Whose Node Is a Leaf FIGURE 25-6 (a) Two possible configurations of a leaf node N; (b) the resulting two possible configurations after removing node © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

33. Removing an Entry Whose Node Has One Child FIGURE 25-7 Removing a node N from its parent node P when N has one child and is itself (a) a left child © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

34. Removing an Entry Whose Node Has One Child FIGURE 25-7 Removing a node N from its parent node P when N has one child and is itself (b) a right child © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

35. Removing an Entry Whose Node Has Two Children FIGURE 25-8 Two possible configurations of a node N that has two children © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

36. Removing an Entry Whose Node Has Two Children FIGURE 25-9 Node N and its subtrees: (a) the entry a is immediately before the entry e, and b is immediately after e; (b) after deleting the node that contained a and replacing e with a © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

37. Removing an Entry Whose Node Has Two Children FIGURE 25-10 The largest entry a in node N’s left subtree occurs in the subtree’s rightmost node R © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

38. Removing an Entry Whose Node Has Two Children FIGURE 25-11 (a) A binary search tree; (b) after removing Chad © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

39. Removing an Entry Whose Node Has Two Children FIGURE 25-11 (c) after removing Sean; (d) after removing Kathy © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

40. Removing an Entry in the Root FIGURE 25-12 (a) Two possible configurations of a root that has one child; (b) after removing the root © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

41. Recursive Implementation Recursive algorithm describes the method’s logic at a high level © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

42. Recursive Implementation The private method removeEntry © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

43. Recursive Implementation The private method removeEntry © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

44. Recursive Implementation The algorithm removeFromRoot © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

45. Recursive Implementation The private method removeFromRoot © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

46. Recursive Implementation The private method removeFromRoot © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

47. Recursive Implementation The private method findLargest © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

48. Recursive Implementation The private method removeLargest © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

49. Iterative Implementation Pseudocode that describes what remove must do: © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

50. Iterative Implementation The public method remove © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

51. Iterative Implementation The public method remove © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

52. Iterative Implementation The private method findNode © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

53. Iterative Implementation Pseudocode for the private method getNodeToRemove © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

54. Iterative Implementation Implementation of the private method getNodeToRemove © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

55. Iterative Implementation The private method removeNode © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

56. Efficiency of Operations For tree of height h  The operations add, remove, and getEntry are O(h) If tree of n nodes has height h = n  These operations are O(n) Shortest tree is full  Results in these operations being O(log n) © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

57. Efficiency of Operations FIGURE 25-13 Two binary search trees that contain the same data © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. O(n) O(log n)

58. The Importance of Balance Do not need a full binary search tree to get O(log n) performance Complete tree will also give us O(log n) performance. © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

59. Order in Which Nodes Are Added Order in which you add entries to a binary search tree affects the shape of the tree If you add entries into an initially empty binary search tree, do not add them in sorted order. © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

60. Implementation of the ADT Dictionary Interface for a dictionary © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

61. Implementation of the ADT Dictionary LISTING 25-3 An outline of an implementation of the ADT dictionary that uses a binary search tree © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

62. Implementation of the ADT Dictionary LISTING 25-3 An outline of an implementation of the ADT dictionary that uses a binary search tree © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

63. Implementation of the ADT Dictionary Dictionary’s add method © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

64. Implementation of the ADT Dictionary Beginning of Dictionary’s remove method © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

65. Implementation of the ADT Dictionary Iterators © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

66. Implementation of the ADT Dictionary Iterators © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

67. Implementation of the ADT Dictionary Iterators © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.

68. End Chapter 25 © 2015 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.