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Trees

Learn how to use trees to organize and process information hierarchically, including recursion, tree traversal, binary search trees, and heaps. Implement these concepts using linked data structures and arrays.

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Trees

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  1. Trees Chapter 8

  2. Chapter Objectives • To learn how to use a tree to represent a hierarchical organization of information • To learn how to use recursion to process trees • To understand the different ways of traversing a tree • To understand the difference between binary trees, binary search trees, and heaps • To learn how to implement binary trees, binary search trees, and heaps using linked data structures and arrays

  3. Chapter Objectives (continued) • To learn how to use a binary search tree to store information so that it can be retrieved in an efficient manner • To learn how to use a Huffman tree to encode characters using fewer bytes than ASCII or Unicode, resulting in smaller files and reduced storage requirements

  4. Tree Terminology • A tree consists of a collection of elements or nodes, with each node linked to its successors • The node at the top of a tree is called its root • The links from a node to its successors are called branches • The successors of a node are called its children • The predecessor of a node is called its parent

  5. Tree Terminology (continued) • Each node in a tree has exactly one parent except for the root node, which has no parent • Nodes that have the same parent are siblings • A node that has no children is called a leaf node • A generalization of the parent-child relationship is the ancestor-descendent relationship

  6. Tree Terminology (continued) • A subtree of a node is a tree whose root is a child of that node • The level of a node is a measure of its distance from the root

  7. Binary Trees • In a binary tree, each node has at most two subtrees • A set of nodes T is a binary tree if either of the following is true • T is empty • Its root node has two subtrees, TL and TR, such that TL and TR are binary trees

  8. Some Types of Binary Trees • Expression tree • Each node contains an operator or an operand • Huffman tree • Represents Huffman codes for characters that might appear in a text file • Huffman code uses different numbers of bits to encode letters as opposed to ASCII or Unicode • Binary search trees • All elements in the left subtree precede those in the right subtree

  9. Some Types of Binary Trees (continued)

  10. Fullness and Completeness • Trees grow from the top down • Each new value is inserted in a new leaf node • A binary tree is full if every node has two children except for the leaves

  11. General Trees • Nodes of a general tree can have any number of subtrees • A general tree can be represented using a binary tree

  12. Tree Traversals • Often we want to determine the nodes of a tree and their relationship • Can do this by walking through the tree in a prescribed order and visiting the nodes as they are encountered • This process is called tree traversal • Three kinds of tree traversal • Inorder • Preorder • Postorder

  13. Tree Traversals (continued) • Preorder: Visit root node, traverse TL, traverse TR • Inorder: Traverse TL, visit root node, traverse TR • Postorder: Traverse TL, Traverse TR, visit root node

  14. Visualizing Tree Traversals • You can visualize a tree traversal by imagining a mouse that walks along the edge of the tree • If the mouse always keeps the tree to the left, it will trace a route known as the Euler tour • Preorder traversal if we record each node as the mouse first encounters it • Inorder if each node is recorded as the mouse returns from traversing its left subtree • Postorder if we record each node as the mouse last encounters it

  15. Visualizing Tree Traversals (continued)

  16. Traversals of Binary Search Trees and Expression Trees • An inorder traversal of a binary search tree results in the nodes being visited in sequence by increasing data value • An inorder traversal of an expression tree inserts parenthesis where they belong (infix form) • A postorder traversal of an expression tree results in postfix form

  17. The Node<E> Class • Just as for a linked list, a node consists of a data part and links to successor nodes • The data part is a reference to type E • A binary tree node must have links to both its left and right subtrees

  18. The BinaryTree<E> Class

  19. The BinaryTree<E> Class (continued)

  20. Overview of a Binary Search Tree • Binary search tree definition • A set of nodes T is a binary search tree if either of the following is true • T is empty • Its root has two subtrees such that each is a binary search tree and the value in the root is greater than all values of the left subtree but less than all values in the right subtree

  21. Overview of a Binary Search Tree (continued)

  22. Searching a Binary Tree

  23. Class TreeSet and Interface Search Tree

  24. BinarySearchTree Class

  25. Insertion into a Binary Search Tree

  26. Removing from a Binary Search Tree

  27. Removing from a Binary Search Tree (continued)

  28. Heaps and Priority Queues • In a heap, the value in a node is les than all values in its two subtrees • A heap is a complete binary tree with the following properties • The value in the root is the smallest item in the tree • Every subtree is a heap

  29. Inserting an Item into a Heap

  30. Removing an Item from a Heap

  31. Implementing a Heap • Because a heap is a complete binary tree, it can be implemented efficiently using an array instead of a linked data structure • First element for storing a reference to the root data • Use next two elements for storing the two children of the root • Use elements with subscripts 3, 4, 5, and 6 for storing the four children of these two nodes and so on

  32. Inserting into a Heap Implemented as an ArrayList

  33. Inserting into a Heap Implemented as an ArrayList (continued)

  34. Priority Queues • The heap is used to implement a special kind of queue called a priority queue • The heap is not very useful as an ADT on its own • Will not create a Heap interface or code a class that implements it • Will incorporate its algorithms when we implement a priority queue class and Heapsort • Sometimes a FIFO queue may not be the best way to implement a waiting line • A priority queue is a data structure in which only the highest-priority item is accessible

  35. Insertion into a Priority Queue

  36. The PriorityQueue Class • Java provides a PriorityQueue<E> class that implements the Queue<E> interface given in Chapter 6. • Peek, poll, and remove methods return the smallest item in the queue rather than the oldest item in the queue.

  37. Design of a KWPriorityQueue Class

  38. Huffman Trees • A Huffman tree can be implemented using a binary tree and a PriorityQueue • A straight binary encoding of an alphabet assigns a unique binary number to each symbol in the alphabet • Unicode for example • The message “go eagles” requires 144 bits in Unicode but only 38 using Huffman coding

  39. Huffman Trees (continued)

  40. Huffman Trees (continued)

  41. Chapter Review • A tree is a recursive, nonlinear data structure that is used to represent data that is organized as a hierarchy • A binary tree is a collection of nodes with three components: a reference to a data object, a reference to a left subtree, and a reference to a right subtree • In a binary tree used for arithmetic expressions, the root node should store the operator that is evaluated last • A binary search tree is a tree in which the data stored in the left subtree of every node is less than the data stored in the root node, and the data stored in the right subtree is greater than the data stored in the root node

  42. Chapter Review (continued) • A heap is a complete binary tree in which the data in each node is less than the data in both its subtrees • Insertion and removal in a heap are both O(log n) • A Huffman tree is a binary tree used to store a code that facilitates file compression

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