Chapter 19 Implementing Trees and Priority Queues Fundamentals of Java.

Chapter 19Implementing Trees and Priority

Queues

Fundamentals of Java

Fundamentals of Java 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.


Objectives (cont.)

Understand the basic tree traversals. Use binary search trees to implement sorted

sets and sorted maps. Use heaps to implement priority queues.


Vocabulary

Binary search tree Binary tree Expression tree General tree Heap Heap property


Vocabulary (cont.)

Interior node Leaf Left subtree Parse tree Right subtree Root


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


An Overview of Trees (cont.)

Figure 19-1: Parse tree for a sentence



Table 19-1: Summary of terms used to describe trees



Table 19-1: Summary of terms used to describe trees (cont.)



Figure 19-2: Tree and some of its properties



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



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 TNode 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)



Figure 19-4: Different types of binary trees



Full binary tree: Contains maximum number of nodes for its height– 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.



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



Expression tree: For evaluating expressions

Figure 19-6: Some expression trees


An Overview of Trees: Binary Search Trees

Figure 19-7: Call tree for the binary search of an array


An Overview of Trees: Binary Search Trees (cont.)

Figure 19-8: Binary search tree



Binary search tree: Each node is greater than or equal to left child and less than or equal to right child.

Recursive search process:



Figure 19-9: Three binary tree shapes with the same data


Binary Tree Traversals

Figure 19-11: Inorder traversal

Figure 19-10: Preorder traversal


Binary Tree Traversals (cont.)

Figure 19-13: Level-order traversal

Figure 19-12: Postorder traversal


Linked Implementation of Binary Trees

Table 19-2: Methods of the BSTPT interface


Linked Implementation of Binary Trees (cont.)

Table 19-2: Methods of the BSTPT interface (cont.)



Figure 19-14: Interfaces and classes used in the binary search tree prototype



Example 19.1: Interface for binary search tree prototypes



Example 19.1: Interface for binary search tree prototypes (cont.)



add method



add method (cont.)



Pseudocode for searching a binary tree:



Inorder traversal code:



Pseudocode for level-order traversal:



Steps for removing a node:



Expanded step 4 for removing a node from a binary tree:


Array Implementation of a Binary Tree

Figure 19-16: Complete binary tree

Figure 19-17: Array representation of a complete binary tree


Array Implementation of a Binary Tree (cont.)

Table 19-3: Locations of given items in an array representation of a complete binary tree


Array Implementation of a Binary Tree (cont.)

Table 19-4: Relatives of a given item in an array representation of a complete binary tree


Implementing Heaps

Table 19-5: Methods in the interface HeapPT


Implementing Heaps (cont.)

add method:


Implementing Heaps (cont.)

pop method:


Implement Heaps (cont.)

pop method (cont.):


Using a Heap to Implement a Priority Queue

Example 19.3: Heap implementation of a priority queue


Using a Heap to Implement a Priority Queue (cont.)

Example 19.3: Heap implementation of a priority queue (cont.)


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.


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.


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.

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Chapter 19 Implementing Trees and Priority Queues Fundamentals of Java.

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