© 2006 Pearson Education Chapter 10: Non-linear Data Structures Presentation slides for Java Software Solutions for AP* Computer Science A 2nd Edition by John Lewis, William Loftus, and Cara Cocking Java Software Solutions is published by Addison-Wesley Presentation slides are copyright 2006 by John Lewis, William Loftus, and Cara Cocking. All rights reserved. Instructors using the textbook may use and modify these slides for pedagogical purposes. *AP is a registered trademark of The College Entrance Examination Board which was not involved in the production of, and does not endorse, this product.

© 2006 Pearson Education 2 Non-linear Data Structures  Now we learn a few more data structures  Chapter 10 focuses on: Sets and maps Trees and binary search trees Heaps Handling collisions in hashtables

© 2006 Pearson Education Sets and Maps  A set is a collection of elements with no duplicates Example: {5, 7, 8} is a set, {3, 3, 4} is not  A map matches, or maps, keys to value Example: a dictionary is a map that maps words to definitions  The keys in a map form a set (and therefore must be unique)  The Set and Map interfaces in Java represent sets and maps  The classes HashSet, TreeSet, HashMap, and TreeMap are implementations

© 2006 Pearson Education Trees and Binary Trees  A tree is a non-linear data structure that consists of zero or more nodes that form a hierarchy  One node is the root node  In a binary tree, each node has at most two children, the right child and left child  Nodes without children are called leaf nodes  A subtree is formed by each child, consisting of the child and its descendents

© 2006 Pearson Education A Tree

© 2006 Pearson Education Binary Tree Implementation  A binary tree is a dynamic data structure  Each node contains data and has references to the left and right children  Basic structure: class TreeNode { Object value; TreeNode left; TreeNode right; }  See TreeNode.java (page 557) TreeNode.java

© 2006 Pearson Education Binary Tree Implementation

© 2006 Pearson Education Tree Traversal  There are 3 ways to traverse a tree, that is, to visit every node:  Preorder traversal: visit the current node, then traverse its left subtree, then its right subtree  Postorder traversal: traverse the left subtree, then the right subtree, then visit the current node  Inorder traversal: traverse the left subtree, then visit the current node, then traverse its right subtree  Tree traversal is a recursive process

© 2006 Pearson Education Binary Search Trees  A binary search tree can be used for storing sorted data  For any node N, every node in N’s left subtree must be less than N, and every node in N’s right subtree must be greater than or equal to N  An inorder traversal of a binary tree visits the nodes in sorted order  See SortGrades.java (page 562) SortGrades.java  See BSTree.java (page 563) BSTree.java  See BSTNode.java (page 565) BSTNode.java

© 2006 Pearson Education A Binary Search Tree

© 2006 Pearson Education Binary Search Trees  Searching for an element is a recursive process  If the desired element is less than the current node, try the left child next  If the desired element is greater than the current node, try the right child next  To insert an element, perform a search to find the proper spot  To delete an element, replace it with its inorder successor

© 2006 Pearson Education Heaps  A heap is a complete binary tree in which each parent has a value less than both its children  A complete binary tree has the maximum number of nodes on every level, except perhaps the bottom, and all the nodes are in the leftmost positions on the bottom  The smallest node in a heap is always at the root

© 2006 Pearson Education Adding to a Heap  To add a node to a heap, add it in the position that keeps the tree complete, then bubble it up if necessary by swapping with its parent until it is not less than its parent

© 2006 Pearson Education Deleting from a Heap  To delete a node from a heap, replace it with the last node on the bottom level and bubble that node up or down as necessary

© 2006 Pearson Education Heapsort  A heap can be used to perform a sort  To do a heapsort, add the elements to a heap, then remove the elements one-by-one by always removing the root node  The nodes will be removed in sorted order  Heapsort is O(n log n)

© 2006 Pearson Education Implementing Heaps  Heaps are often implemented using lists since it is more convenient to add and delete items  The children of the i th element are the 2i th and 2i+1 th elements in the list

© 2006 Pearson Education Hashtables  We can handle collisions in hashtables using chaining or open addressing techniques  With chaining, each cell in the hashtable is a linked list and thus many elements can be stored in the same cell

© 2006 Pearson Education Hashtables  With open addressing techniques, when a collision occurs, a new hash code is calculated, called rehashing  Rehashing continues until a free cell is found  Linear probing, a simple rehash method, probes down the hashtable (wrapping around when the end is reached) until a free cell is found  See Guests.java (page 586) Guests.java

© 2006 Pearson Education Summary  Chapter 10 has focused on: Sets and maps Trees and binary search trees Heaps Handling collisions in hashtables