Chapter Contents. Trees. Tree Concepts. Hierarchical Organization. Hierarchical Organization. Hierarchical Organization.

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1 Chapter Contents Chapter 18 Tree Concepts Hierarchical Organizations Tree Terminology Traversals of a Tree Traversals of a Binary Tree Traversals of a General Tree Java Interfaces for Interfaces for All Interface for Binary Examples of Binary Expression Decision Binary Search Examples of General Parse Game 2 Tree Concepts Previous data organizations place data in linear order Some data organizations require categorizing data into groups, subgroups This is hierarchical classification Data items appear at various levels within the organization Hierarchical Organization Example: File directories 3 Computer files organized into folders. 4 Hierarchical Organization Example: A university's organization Hierarchical Organization Example: Family trees A university's administrative structure. 5 Carole's children and grandchildren. 6 1

2 Tree Terminology A tree is A set of nodes Connected by edges The edges indicate relationships among nodes Nodes arranged in levels Indicate the nodes' hierarchy Top level is a single node called the root 7 Tree Terminology A tree equivalent to the my stuff tree 8 Tree Terminology Tree Terminology Nodes at a given level are children of nodes of previous level Node with children is the parent node of those children Nodes with same parent are siblings Node with no children is a leaf node The only node with no parent is the root node Empty trees? Some authors specify a general tree must have at least the root node This text will allow all trees to be empty A node is reached from the root by a path The length of the path is the number of edges that compose it The height of a tree is the number of levels in the tree The subtree of a node is a tree rooted at a child of that node All others have one parent each 9 10 Binary Each node has at most two children Binary A binary tree is either empty or has the following form Where T left and T right are binary trees Three binary trees

3 Binary Every nonleaf in a full binary tree has exactly two children A complete binary tree is full to its next-tolast level Leaves on last level filled from left to right The height of a binary tree with n nodes that is either complete or full is log 2 (n + 1) Binary The number of nodes in a full binary tree as a function of the tree's height Traversals of a Tree Traversals of a Tree Visiting a node Processing the data within a node This is the action performed on each node during traversal of a tree A traversal can pass through a node without visiting it at that moment For a binary tree Visit the root Visit all nodes in the root's left subtree Preorder traversal: visit root before the subtrees The visitation order of a preorder traversal. Visit all nodes in the root's right subtree Traversals of a Tree Inorder traversal: visit root between visiting the subtrees Traversals of a Tree Postorder traversal: visit root after visiting the subtrees These These are are examples of of a depth-first traversal. The visitation order of an inorder traversal. 17 The visitation order of a postorder traversal. 18 3

4 Traversals of a Tree Traversals Exercise Level-order traversal: begin at the root, visit nodes one level at a time This This is is an an example of of a breadth-first traversal. The visitation order of a level-order traversal. 19 The order of these nodes being visited using 4 different traversal methods 20 Answer Traversals of a General Tree In this binary tree, D = node, L = left, R = right Preorder (DLR) traversal yields: A, H, G, I, F, E, B, C, D Postorder (LRD) traversal yields: G, F, E, I, H, D, C, B, A In-order (LDR) traversal yields: G, H, F, I, E, A, B, D, C Level-order traversal yields: A, H, B, G, I, A general tree has traversals that are in Level order Preorder Postorder Inorder traversal not well defined for a general tree C, F, E, D Traversals of a General Tree The visitation order of two traversals of a general tree: (a) preorder; (b) postorder. 23 Java Interfaces for An interface that specifies operations common to all trees public interface TreeInterface { public Object getrootdata(); public int getheight(); public int getnumberofnodes(); public boolean isempty(); public void clear(); } // end TreeInterface 24 4

5 Java Interfaces for Interface for iterators for various traversals Java Interfaces for Interface for a class of binary trees import java.util.iterator; public interface TreeIteratorInterface { public Iterator getpreorderiterator(); public Iterator getpostorderiterator(); public Iterator getinorderiterator(); public Iterator getlevelorderiterator(); } // end TreeIteratorInterface 25 public interface BinaryTreeInterface extends TreeInterface, TreeIteratorInterface { /** Sets an existing binary tree to a new one-node binary tree. rootdata an object that is the data in the new tree s root */ public void settree(object rootdata); /** Sets an existing binary tree to a new binary tree. rootdata an object that is the data in the new tree s root lefttree the left subtree of the new tree righttree the right subtree of the new tree */ public void settree(object rootdata, BinaryTreeInterface lefttree, BinaryTreeInterface righttree); } // end BinaryTreeInterface 26 Construct A Binary Tree BinaryTreeInterface<String> ftree; ftree = new BinaryTree<String>(); dtree.settree( D ); gtree = new BinaryTree<String>(); gtree.settree( G ); etree = new BinaryTree<String>(); etree.settree( E, ftree, gtree);.. Expression The root contains the binary operator and children contain the operands. Order of children matches the order of operands. Such binary tree is called expression. 27 Expression trees for four algebraic expressions.. 28 Expression Inorder traversal produces the original infix expression, but without any parentheses Preorder traversal produces the prefix expression Postorder traversal produces the postfix expression (b) preorder: +*abc postorder: ab*c+ 29 Expression Algorithm for evaluating an expression tree in postorder traversal Algorithm evaluate(expressiontree) if (expressiontree is empty) return 0 else { firstoperand = evaluate(left subtree of expressiontree) secondoperand = evaluate(right subtree of expressiontree) operator = the root of expressiontree return the result of the operation operator and its operands firstoperand and secondoperand } 30 5

6 Exercises Draw an expression tree for each of these algebraic expressions. a. a+b*c b. (a+b)*c 31 Decision Basis of Expert System An expert system also known as a knowledge based system that contains some knowledge. It was first developed by researchers in artificial intelligence. The most common form of expert systems is made up of a set of rules that analyze information. Each parent in a decision tree is a question that has a finite number of response. Each possible answer to the question corresponds to a child of the node. Leave node are conclusions that have no children 32 Decision Helps users solve problems, make decisions Decision A possible Java interface for a binary decision tree. A binary decision tree. 33 public interface DecisionTreeInterface extends BinaryTreeInterface { /** Task: Gets the data in the current node. the data object in the current node */ public Object getcurrentdata(); /** Task: Determines whether current node contains an answer. true if the current node is a leaf */ public boolean isanswer(); /** Task: Moves the current node to the left (right) child of the current node. */ public void advancetono(); public void advancetoyes(); /** Task: Sets the current node to the root of the tree.*/ public void reset(); } // end DecisionTreeInterface 34 Decision The decision tree for a guessing game Binary Search A search tree organizes its data so that a search is more efficient Binary search tree Nodes contain Comparable objects A node's data is greater than the data in the node's left subtree A node's data is less than the data in the node's right subtree Facts are used to build this decision tree

7 Binary Search Binary Search A binary search tree of names. 37 Two binary search trees containing the same names as the tree in previous slide 38 Questions Answer How many different binary search trees can you form from the strings a, b, and c? What are the heights of the shortest and tallest trees that you formed? How many different binary search trees can you form from the strings a, b, and c? 5 What are the heights of the shortest and tallest trees that you formed? 2 and Binary Search Binary Search Search begins at the root, ends at either the target node or leaf node. The number of comparison is the number of nodes along the path from the root to target node. The height of a tree directly affects the length of the longest path from the root to a leaf and hence affects the efficiency of a worst-case search. Searching a binary search tree of height h is An algorithm for searching a binary search tree Algorithm bstsearch(binarysearchtree, desiredobject) // Searches a binary search tree for a given object. // Returns true if the object is found. if (binarysearchtree is empty) return false else if (desiredobject == object in the root of binarysearchtree) return true else if (desiredobject < object in the root of binarysearchtree) return bstsearch(left subtree of binarysearchtree, desiredobject) else return bstsearch(right subtree of binarysearchtree, desiredobject) O(h)

8 Heaps A complete binary tree Nodes contain Comparable objects Each node contains no smaller (or no larger) than objects in its descendants Maxheap Object in a node is its descendant objects. Root node contains the largest data Minheap Object in a node is descendant objects Root node contains the smallest data 43 Heaps (a) A maxheap and (b) a minheap that contain the same values 44 Parse Tree An algebraic expression is either a term or two terms separated by + or operator A term is either a factor or two factors separated by * or / operator. A factor is either a variable or an algebraic expression enclosed in parentheses. A variable is a single letter These rules form a grammar for algebraic expression, much like English language A parse tree for the algebraic expression a * (b + c) Parse Tree grammar Exercise Answer Draw a parse tree for the algebraic expression a*b + c <term> <Expression> + <term> <factor> * <factor> <factor> <Variable> <Variable> <factor> 47 a b C 48 8

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