COSC2100: Data Structures and Algorithms Tree Basics and Binary Tree

Review So far, we have learnt two fundamental data storage structures Array Linked List An ordered array Search a key in the array: O(lgN) with binary search Insert or delete a new key: O(N) worst case Linked list Search a key in the list: O(N) worst case Insert a key: O(1)

Overview We are going to learn another fundamental data storage structure Binary Tree With a special binary tree, Balanced Binary Search Tree Search: O(lgN) Insert and Delete: O(lgN) The basics about trees Terminology Tree traversals

Tree Structure in Real Life College College of Art and Science Departments BSCI MSCS CHEM Course types MATH COSC COSC courses COSC1000 COSC2100 COSC3100 Other examples: Family tree Table of contents Company organization chart

A file system that stores documentations for several courses Trees 11/10/2018 5:42 PM Trees in Computers /usr/rt/courses cosc1020 cosc2100 cosc3100 programs homework lectures A file system that stores documentations for several courses Trees

Tree Data Structure A tree consists of nodes connected by edges Trees 11/10/2018 5:42 PM Tree Data Structure A tree consists of nodes connected by edges Nodes are represented as circles in the graph Edges are lines in the graph Trees are hierarchical structures B C A D E G H X Y Z Trees

Tree Terminology I Root: the node at the top of the tree Trees 11/10/2018 5:42 PM Tree Terminology I A B D C G H E F I J K Root: the node at the top of the tree Node A Edge: directed connection from a node’s parent to the node Edges from A to B, from B to F, from F to K Parent: the node to which the given node is connected via the upward edge B’s parent: A A’s parent: null Child: the node to which the given node is connected via downward edges A’s child: B, C, D C’s child: G, H Trees

Tree Terminology II Internal node: node with at least one child Trees 11/10/2018 5:42 PM Tree Terminology II Internal node: node with at least one child Nodes A, B, C, F leaf node node without children Nodes E, I, J, K, G, H, D Siblings of a node The nodes with the same parent I’s siblings: J and K Subtree (CGH): tree consisting of a node and its children, children’s children… Root of the subtree: C Edge from A (C’s parent) to C subtree A B D C G H E F I J K Trees

Trees 11/10/2018 5:42 PM Tree Terminology III Path: a path form node n1 to nk is defined as a sequence of nodes n1, n2, …, nk such that ni is the parent of ni+1 for 1 <= i < k Length: the length of this path is the number of edges on the path The depth of a node ni is the length of path form the root to ni. The depth of the root is 0 The depth of the tree is the largest depth among leaves. The height of ni is the length of the longest path from ni to a leaf. All leaves are at height 0. The height of a tree is the height of the root. Trees

Examples Exercises: Parent of J: children of J: B D C G H E F I J K Exercises: Parent of J: children of J: Path from A to F and its length: Height and depth of F:

More Exercises Which node is the root? Which nodes are leaves? The tree’s depth? For node E, list its parent children siblings height depth B C D E F G H I J K L M

Yet More Exercises Which node is the root? Which nodes are leaves? The tree’s depth? For node A, list its parent children siblings height depth A

Trees 11/10/2018 5:42 PM Binary Trees A A binary tree is a tree in which every node has at most two children left child and right child A node in a binary tree doesn’t necessarily have two children only a left child only a right child or no children at all B C D E G H I Trees

Representing Tree in Java Code Trees 11/10/2018 5:42 PM Representing Tree in Java Code class Node { int iData; Node left; Node right; //constructor //display } class Tree { private Node root; //constructor //methods for insertion, deletion… } Node is the building blocks for Tree, just as Link for LinkList Node root is the only entry to a Tree object, as Link first to a LinkList object Trees

Traversing a Tree To traverse a tree means to visit all the nodes in some specified order Inorder traversal Preorder traversal Postorder traversal

Trees 11/10/2018 5:42 PM Inorder Traversal Algorithm inOrder(v) If v==null Return inOrder (left (v)) System.out.print(v.iData + “ “) inOrder (right (v)) A node is visited after its left subtree and before its right subtree What is printed for the given tree if executing inOrder(root) 6 2 8 1 4 7 9 3 5 Trees

Preorder Traversal A node is visited before its children Trees 11/10/2018 5:42 PM Preorder Traversal A node is visited before its children If visiting a node is printing the string, what will be printed for the given tree? Algorithm preOrder(v) visit(v) for each child w of v preorder (w) 1 Make Money Fast! 2 5 1. Motivations 2. Methods 7 6 3 4 2.1 Stock Fraud 2.3 Bank Robbery 1.1 Greed 1.2 Avidity Trees

Postorder Traversal A node is visited after its descendants Trees 11/10/2018 5:42 PM Postorder Traversal A node is visited after its descendants Application: compute space used by files in a directory and its subdirectories Demonstrate how it works. Algorithm postOrder(v) for each child w of v postOrder (w) visit(v) 7 cs16/ 3 6 homeworks/ programs/ 1 2 4 5 h1c.doc 3K h1nc.doc 2K DDR.java 10K Robot.java 20K Trees

Pop Quiz A By visiting the node, we print the character stored in the node Show the result of Preorder Inorder Postorder of the given tree. B C D E F G H I J K L M

Summary Tree Terminologies Tree traversals Hierarchical data structure Root Internal, external (leaf) nodes Parent Depth Height Path Tree traversals Preorder, inorder, and postorder