1. Chapter 19 Implementing Trees and Priority Queues Fundamentals of Java

2. 2 Objectives Use the appropriate terminology to describe trees. Distinguish different types of hierarchical collections such as general trees, binary trees, binary search trees, and heaps.

3. Fundamentals of Java 3 Objectives (cont.) Understand the basic tree traversals. Use binary search trees to implement sorted sets and sorted maps. Use heaps to implement priority queues.

4. Fundamentals of Java 4 Vocabulary Binary search tree Binary tree Expression tree General tree Heap Heap property

5. Fundamentals of Java 5 Vocabulary (cont.) Interior node Leaf Left subtree Parse tree Right subtree Root

6. Fundamentals of Java 6 An Overview of Trees Tree: Data structure in which each item can have multiple successors – All items have exactly one predecessor. Except a privileged item called the root Parse tree: Describes the syntactic structure of a sentence in terms of its component parts – Noun phrases and verb phrases

7. Fundamentals of Java 7 An Overview of Trees (cont.) Figure 19-1: Parse tree for a sentence

8. Fundamentals of Java 8 An Overview of Trees (cont.) Table 19-1: Summary of terms used to describe trees

9. Fundamentals of Java 9 An Overview of Trees (cont.) Table 19-1: Summary of terms used to describe trees (cont.)

10. Fundamentals of Java 10 An Overview of Trees (cont.) Figure 19-2: Tree and some of its properties

11. Fundamentals of Java 11 An Overview of Trees (cont.) General trees: Trees with no restrictions on number of children Binary trees: Each node has at most two children: left child and right child. Figure 19-3: Two unequal binary trees that have equal sets of nodes

12. Fundamentals of Java 12 An Overview of Trees (cont.) Recursive processing of trees is common, so useful to have recursive definitions of trees – General tree: Either empty or consists of a finite set of nodes T Node r is the root. Set T - {r} partitioned into disjoint subsets (general trees) – Binary tree: Either empty or consists of a root plus a left subtree and a right subtree (binary trees)

13. Fundamentals of Java 13 An Overview of Trees (cont.) Figure 19-4: Different types of binary trees

14. Fundamentals of Java 14 An Overview of Trees (cont.) Full binary tree: Contains maximum number of nodes for its height – Fully balanced – If height is d, 2 d -1 nodes – Level n has up to 2 n nodes. – Height of a fully balanced tree of n nodes is log 2 n.

15. Fundamentals of Java 15 An Overview of Trees (cont.) Heap: Binary tree in which the item in each node is less than or equal to the items in both of its children – Heap property Figure 19-5: Examples of heaps

16. Fundamentals of Java 16 An Overview of Trees (cont.) Expression tree: For evaluating expressions Figure 19-6: Some expression trees

17. Fundamentals of Java 17 An Overview of Trees: Binary Search Trees Figure 19-7: Call tree for the binary search of an array

18. Fundamentals of Java 18 An Overview of Trees: Binary Search Trees (cont.) Figure 19-8: Binary search tree

19. Fundamentals of Java 19 An Overview of Trees: Binary Search Trees (cont.) Binary search tree: Each node is greater than or equal to left child and less than or equal to right child. Recursive search process:

20. Fundamentals of Java 20 An Overview of Trees: Binary Search Trees (cont.) Figure 19-9: Three binary tree shapes with the same data

21. Fundamentals of Java 21 Binary Tree Traversals Figure 19-11: Inorder traversal Figure 19-10: Preorder traversal

22. Fundamentals of Java 22 Binary Tree Traversals (cont.) Figure 19-13: Level-order traversal Figure 19-12: Postorder traversal

23. Fundamentals of Java 23 Linked Implementation of Binary Trees Table 19-2: Methods of the BSTPT interface

24. Fundamentals of Java 24 Linked Implementation of Binary Trees (cont.) Table 19-2: Methods of the BSTPT interface (cont.)

25. Fundamentals of Java 25 Linked Implementation of Binary Trees (cont.) Figure 19-14: Interfaces and classes used in the binary search tree prototype

26. Fundamentals of Java 26 Linked Implementation of Binary Trees (cont.) Example 19.1: Interface for binary search tree prototypes

27. Fundamentals of Java 27 Linked Implementation of Binary Trees (cont.) Example 19.1: Interface for binary search tree prototypes (cont.)

28. Fundamentals of Java 28 Linked Implementation of Binary Trees (cont.) add method

29. Fundamentals of Java 29 Linked Implementation of Binary Trees (cont.) add method (cont.)

30. Fundamentals of Java 30 Linked Implementation of Binary Trees (cont.) Pseudocode for searching a binary tree:

31. Fundamentals of Java 31 Linked Implementation of Binary Trees (cont.) Inorder traversal code:

32. Fundamentals of Java 32 Linked Implementation of Binary Trees (cont.) Pseudocode for level-order traversal:

33. Fundamentals of Java 33 Linked Implementation of Binary Trees (cont.) Steps for removing a node:

34. Fundamentals of Java 34 Linked Implementation of Binary Trees (cont.) Expanded step 4 for removing a node from a binary tree:

35. Fundamentals of Java 35 Array Implementation of a Binary Tree Figure 19-16: Complete binary tree Figure 19-17: Array representation of a complete binary tree

36. Fundamentals of Java 36 Array Implementation of a Binary Tree (cont.) Table 19-3: Locations of given items in an array representation of a complete binary tree

37. Fundamentals of Java 37 Array Implementation of a Binary Tree (cont.) Table 19-4: Relatives of a given item in an array representation of a complete binary tree

38. Fundamentals of Java 38 Implementing Heaps Table 19-5: Methods in the interface HeapPT

39. Fundamentals of Java 39 Implementing Heaps (cont.) add method:

40. Fundamentals of Java 40 Implementing Heaps (cont.) pop method:

41. Fundamentals of Java 41 Implement Heaps (cont.) pop method (cont.):

42. Fundamentals of Java 42 Using a Heap to Implement a Priority Queue Example 19.3: Heap implementation of a priority queue

43. Fundamentals of Java 43 Using a Heap to Implement a Priority Queue (cont.) Example 19.3: Heap implementation of a priority queue (cont.)

44. Fundamentals of Java 44 Summary There are various types of trees or hierarchical collections such as general trees, binary trees, binary search trees, and heaps. The terminology used to describe hierarchical collections is borrowed from biology, genealogy, and geology.

45. Fundamentals of Java 45 Summary (cont.) Tree traversals: preorder, inorder, postorder, and level-order traversal A binary search tree preserves a natural ordering among its items and can support operations that run in logarithmic time. Binary search trees are useful for implementing sorted sets and sorted maps.

46. Fundamentals of Java 46 Summary (cont.) Heap – Useful for ordering items according to priority – Guarantees logarithmic insertions and removals – Useful for implementing priority queues Binary search trees typically have a linked implementation. Heaps typically have an array representation.