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1 Presentation for use with the textbook ata Structures and lgorithms in Java, 6 th edition, by M. T. Goodrich, R. Tamassia, and M. H. Goldwasser, Wiley, 014 Trees Mammal og Pig at Trees 1 What is a Tree In computer science, a tree is an abstract model omputers R Us of a hierarchical structure tree consists of nodes Sales Manufacturing with a parent-child relation pplications: US International Laptops esktops n Organization charts n File systems n Programming Europe sia anada environments R& Trees 1
2 Tree Terminology Root: node without parent () Internal node: node with at least one child (,,, F) External node (a.k.a. leaf ): node without children (E, I, J, K, G, H, ) ncestors of a node: parent, grandparent, grand-grandparent, etc. epth of a node: number of ancestors E Height of a tree: maximum depth F of any node (3) escendant of a node: child, grandchild, grand-grandchild, etc. I J K Subtree: tree consisting of a node and its descendants G H subtree Trees 3 Tree T We use positions to abstract nodes Generic methods: n integer size() n boolean isempty() n Iterator iterator() n Iterable positions() ccessor methods: n position root() n position parent(p) n Iterable children(p) n Integer numhildren(p) " Query methods: n boolean isinternal(p) n boolean isexternal(p) n boolean isroot(p) " dditional update methods may be defined by data structures implementing the Tree T Trees 4
3 Java Interface Methods for a Tree interface: Trees 5 Preorder Traversal traversal visits the nodes of a tree in a systematic manner In a preorder traversal, a node is visited before its descendants pplication: print a structured document 1 Make Money Fast! lgorithm preorder(v) visit(v) for each child w of v preorder (w) Motivations. Methods References Greed 1. vidity Stock Fraud. Ponzi Scheme.3 ank Robbery Trees 6 3
4 Postorder Traversal In a postorder traversal, a node is visited after its descendants pplication: compute space used by files in a directory and its subdirectories 9 cs16/ 1 h1c.doc 3K 3 homeworks/ h1nc.doc K R.java 10K lgorithm postorder(v) for each child w of v postorder (w) visit(v) 7 programs/ Stocks.java 5K Robot.java 0K 8 todo.txt 1K Trees 7 inary Trees binary tree is a tree with the following properties: n Each internal node has at most two children (exactly two for proper binary trees) n The children of a node are an ordered pair We call the children of an internal node left child and right child lternative recursive definition: a binary tree is either n a tree consisting of a single node, or n a tree whose root has an ordered pair of children, each of which is a binary tree pplications: n arithmetic expressions n decision processes n searching E F G H I Trees 8 4
5 rithmetic Expression Tree inary tree associated with an arithmetic expression n internal nodes: operators n external nodes: operands Example: arithmetic expression tree for the expression ( (a - 1) + (3 b)) + - a 1 3 b Trees 9 ecision Tree inary tree associated with a decision process n internal nodes: questions with yes/no answer n external nodes: decisions Example: dining decision Yes Want a fast meal? No How about coffee? On expense account? Yes No Yes No Starbucks hipotle Gracie s afé Paragon Trees 10 5
6 Properties of Proper inary Trees Notation n number of nodes e number of external nodes i number of internal nodes h height " Properties: n e = i + 1 n n = e - 1 n h i n h (n - 1)/ n e h n h log e n h log (n + 1) - 1 Trees 11 inarytree T The inarytree T extends the Tree T, i.e., it inherits all the methods of the Tree T dditional methods: n position left(p) n position right(p) n position sibling(p) The above methods return null when there is no left, right, or sibling of p, respectively Update methods may be defined by data structures implementing the inarytree T Trees 1 6
7 Inorder Traversal In an inorder traversal a node is visited after its left subtree and before its right subtree pplication: draw a binary tree n x(v) = inorder rank of v n y(v) = depth of v 6 lgorithm inorder(v) if left (v) null inorder (left (v)) visit(v) if right(v) null inorder (right (v)) Trees 13 Print rithmetic Expressions Specialization of an inorder traversal n print operand or operator when visiting node n print ( before traversing left subtree n print ) after traversing right subtree + lgorithm printexpression(v) if left (v) null print( ( ) inorder (left(v)) print(v.element ()) if right(v) null inorder (right(v)) print ( ) ) - a 1 3 b (( (a - 1)) + (3 b)) Trees 14 7
8 Evaluate rithmetic Expressions Specialization of a postorder traversal n recursive method returning the value of a subtree n when visiting an internal node, combine the values of the subtrees + lgorithm evalexpr(v) if isexternal (v) return v.element () else x evalexpr(left(v)) y evalexpr(right(v)) operator stored at v return x y Trees 15 Euler Tour Traversal Generic traversal of a binary tree Includes a special cases the preorder, postorder and inorder traversals Walk around the tree and visit each node three times: n on the left (preorder) n from below (inorder) n on the right (postorder) + L R Trees 16 8
9 Linked Structure for Trees node is represented by an object storing n Element n Parent node n Sequence of children nodes Node objects implement the Position T F F E E Trees 17 Linked Structure for inary Trees node is represented by an object storing n Element n Parent node n Left child node n Right child node Node objects implement the Position T E E Trees 18 9
10 rray-ased Representation of inary Trees Nodes are stored in an array 0 G H Node v is stored at [rank(v)] n rank(root) = 0 n if node is the left child of parent(node), rank(node) = rank(parent(node)) + 1 n if node is the right child of parent(node), rank(node) = rank(parent(node)) E 9 10 G F H 5 6 J Trees 19 10
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