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Chapter 19 Implementing Trees and Priority Queues Fundamentals of Java Objectives 2 Use the appropriate terminology to describe trees. Distinguish different types of hierarchical collections such as general trees, binary trees, binary search trees, and heaps. Fundamentals of Java Objectives (cont.) 3 Understand the basic tree traversals. Use binary search trees to implement sorted sets and sorted maps. Use heaps to implement priority queues. Fundamentals of Java Vocabulary 4 Binary search tree Binary tree Expression tree General tree Heap Heap property Fundamentals of Java Vocabulary (cont.) 5 Interior node Leaf Left subtree Parse tree Right subtree Root Fundamentals of Java An Overview of Trees Tree: Data structure in which each item can have multiple successors – All items have exactly one predecessor. Except Parse tree: Describes the syntactic structure of a sentence in terms of its component parts – 6 a privileged item called the root Noun phrases and verb phrases Fundamentals of Java An Overview of Trees (cont.) Figure 19-1: Parse tree for a sentence 7 Fundamentals of Java An Overview of Trees (cont.) Table 19-1: Summary of terms used to describe trees 8 Fundamentals of Java An Overview of Trees (cont.) 9 Table 19-1: Summary of terms used to describe trees (cont.) Fundamentals of Java An Overview of Trees (cont.) Figure 19-2: Tree and some of its properties 10 Fundamentals of Java An Overview of Trees (cont.) 11 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 Fundamentals of Java 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) – 12 Binary tree: Either empty or consists of a root plus a left subtree and a right subtree (binary trees) Fundamentals of Java An Overview of Trees (cont.) Figure 19-4: Different types of binary trees 13 Fundamentals of Java An Overview of Trees (cont.) Full binary tree: Contains maximum number of nodes for its height – – – – 14 Fully balanced If height is d, 2d-1 nodes Level n has up to 2n nodes. Height of a fully balanced tree of n nodes is log2n. Fundamentals of Java 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 15 Fundamentals of Java An Overview of Trees (cont.) Expression tree: For evaluating expressions Figure 19-6: Some expression trees 16 Fundamentals of Java An Overview of Trees: Binary Search Trees Figure 19-7: Call tree for the binary search of an array 17 Fundamentals of Java An Overview of Trees: Binary Search Trees (cont.) Figure 19-8: Binary search tree 18 Fundamentals of Java An Overview of Trees: Binary Search Trees (cont.) 19 Binary search tree: Each node is greater than or equal to left child and less than or equal to right child. Recursive search process: Fundamentals of Java An Overview of Trees: Binary Search Trees (cont.) Figure 19-9: Three binary tree shapes with the same data 20 Fundamentals of Java Binary Tree Traversals Figure 19-10: Preorder traversal Figure 19-11: Inorder traversal 21 Fundamentals of Java Binary Tree Traversals (cont.) Figure 19-12: Postorder traversal 22 Figure 19-13: Level-order traversal Fundamentals of Java Linked Implementation of Binary Trees 23 Table 19-2: Methods of the BSTPT interface Fundamentals of Java Linked Implementation of Binary Trees (cont.) Table 19-2: Methods of the BSTPT interface (cont.) 24 Fundamentals of Java Linked Implementation of Binary Trees (cont.) Figure 19-14: Interfaces and classes used in the binary search tree prototype 25 Fundamentals of Java Linked Implementation of Binary Trees (cont.) 26 Example 19.1: Interface for binary search tree prototypes Fundamentals of Java Linked Implementation of Binary Trees (cont.) 27 Example 19.1: Interface for binary search tree prototypes (cont.) Fundamentals of Java Linked Implementation of Binary Trees (cont.) 28 add method Fundamentals of Java Linked Implementation of Binary Trees (cont.) 29 add method (cont.) Fundamentals of Java Linked Implementation of Binary Trees (cont.) 30 Pseudocode for searching a binary tree: Fundamentals of Java Linked Implementation of Binary Trees (cont.) 31 Inorder traversal code: Fundamentals of Java Linked Implementation of Binary Trees (cont.) 32 Pseudocode for level-order traversal: Fundamentals of Java Linked Implementation of Binary Trees (cont.) 33 Steps for removing a node: Fundamentals of Java Linked Implementation of Binary Trees (cont.) 34 Expanded step 4 for removing a node from a binary tree: Fundamentals of Java Array Implementation of a Binary Tree Figure 19-16: Complete binary tree Figure 19-17: Array representation of a complete binary tree 35 Fundamentals of Java Array Implementation of a Binary Tree (cont.) Table 19-3: Locations of given items in an array representation of a complete binary tree 36 Fundamentals of Java Array Implementation of a Binary Tree (cont.) Table 19-4: Relatives of a given item in an array representation of a complete binary tree 37 Fundamentals of Java Implementing Heaps Table 19-5: Methods in the interface HeapPT 38 Fundamentals of Java Implementing Heaps (cont.) 39 add method: Fundamentals of Java Implementing Heaps (cont.) 40 pop method: Fundamentals of Java Implement Heaps (cont.) 41 pop method (cont.): Fundamentals of Java Using a Heap to Implement a Priority Queue Example 19.3: Heap implementation of a priority queue 42 Fundamentals of Java Using a Heap to Implement a Priority Queue (cont.) 43 Example 19.3: Heap implementation of a priority queue (cont.) Fundamentals of Java Summary 44 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. Fundamentals of Java Summary (cont.) 45 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. Fundamentals of Java Summary (cont.) Heap – – – 46 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. Fundamentals of Java