Trees

1. Trees CSCI 3333 Data Structures byDr. Bun YueProfessor of Computer Scienceyue@uhcl.eduhttp://sce.uhcl.edu/yue/2013

2. Acknowledgement • Mr. Charles Moen • Dr. Wei Ding • Ms. Krishani Abeysekera • Dr. Michael Goodrich

3. Computers”R”Us Sales Manufacturing R&D US International Laptops Desktops Europe Asia Canada What is a Tree • In computer science, a tree is an abstract model of a hierarchical structure • A tree consists of nodes with a parent-child relation

4. Trees • Each node in the tree has • 0 or 1 parent node. • 0 or more child nodes. • Only the root node has no parent. • A tree can be considered as a special case of a directed (acyclic) graph. • Example: a family tree may not necessarily be a tree in the computing sense.

5. Trees British Royal Family Tree Queen Victoria King Edward VII Alice Albert Victor King George V King Edward VIII King George VI Queen Elizabeth II Charles Ann Andrew Edward Beatrice Eugenie William Henry Peter Zara

6. Some Tree Applications • Applications: • Organization charts • File Folders • Email Repository • Programming environments: e.g. drop down menus.

7. Recursive Definition of Trees • An empty tree is a tree. • A tree contains a root node connected to 0 or more child sub-trees.

8. Trees Are these trees? 1 2 4 3

9. A C D B E G H F K I J Tree Terminology • Subtree: tree consisting of a node and its descendants • Root: node without parent (A) • Internal node: node with at least one child (A, B, C, F) • External node (a.k.a. leaf ): node without children (E, I, J, K, G, H, D) • Ancestors of a node: parent, grandparent, grand-grandparent, etc. • Depth of a node: number of ancestors • Height of a tree: maximum depth of any node (3) • Descendant of a node: child, grandchild, grand-grandchild, etc. subtree

10. Tree ADT (§ 6.1.2) • Query methods: • boolean isInternal(p) • boolean isExternal(p) • boolean isRoot(p) • Update method: • object replace (p, o) • Additional update methods may be defined by data structures implementing the Tree ADT • We use positions to abstract nodes • Generic methods: • integer size() • boolean isEmpty() • Iterator elements() • Iterator positions() • Accessor methods: • position root() • position parent(p) • positionIterator children(p)

11. Interface TreeNode in Java • Enumeration children(): Returns the children of the receiver as an Enumeration. • boolean getAllowsChildren(): Returns true if the receiver allows children. • TreeNode getChildAt(int childIndex): Returns the child TreeNode at index childIndex. • int getChildCount(): Returns the number of children TreeNodes the receiver contains. • int getIndex(TreeNode node): Returns the index of node in the receivers children. • TreeNode getParent(): Returns the parent TreeNode of the receiver. • Boolean isLeaf(): Returns true if the receiver is a leaf.

12. Interface MutatableTreeNode • Sub-interface of TreeNode • Methods: • void insert(MutableTreeNode child, int index): Adds child to the receiver at index. • void remove(int index): Removes the child at index from the receiver. • void remove(MutableTreeNode node): Removes node from the receiver. • void removeFromParent(): Removes the receiver from its parent. • void setParent(MutableTreeNode newParent): Sets the parent of the receiver to newParent. • void setUserObject(Object object): Resets the user object of the receiver to object.

13. Size of a tree • Size = number of nodes in the tree. • Recursion analysis: • Base case: empty root => return 0; • Recursive case: non-empty root => return 1 + sum of size of all child sub-trees.

14. Size of a tree Algorithm size(TreeNode root) Input TreeNode root Output: size of the tree if (root = null) return 0; else return 1 + Sum of sizes of all children of root end if

15. Implementation in Java public static int size(TreeNode root) { if (root == null) return 0; Enumeration enum = root.children(); int result = 1; // count the root. while (enum.hasMoreElement()) { result += size((TreeNode) enum.nextElement()); } return result; }

16. Trees in languages • There are no general tree classes in the standard libraries in many languages: e.g. C++ STL, Ruby, Python, etc. • No single interface tree design satisfies all users. • There may be specialized trees, especially for GUI design. E.g. • JTree in Java • FXTreeList in Ruby

17. Tree Traversal • Traversing is the systematic way of accessing, or ‘visiting’ all the nodes in a tree. • Consider the three operations on a binary tree: • V: Visit a node • L: (Recursively) traverse the left subtree of a node • R: (Recursively) traverse the right subtree of a node

18. Tree Traversing • We can traverse a tree in six different orders: LVR VLR LRV VRL RVL RLV

19. Preorder Traversal AlgorithmpreOrder(v) visit(v) foreachchild w of v preorder (w) • A traversal visits the nodes of a tree in a systematic manner • In a preorder traversal, a node is visited before its descendants • Application: print a structured document 1 Make Money Fast! 2 5 9 1. Motivations 2. Methods References 6 7 8 3 4 2.3 BankRobbery 2.1 StockFraud 2.2 PonziScheme 1.1 Greed 1.2 Avidity

20. Java Implementation public static void preorder(TreeNode root) { if (root != null) { visit(root); // assume declared. Enumernation enum = root.children(); while (enum.hasMoreElement()) { preorder((TreeNode) enum.nextElement()); } }

21. Postorder Traversal AlgorithmpostOrder(v) foreachchild w of v postOrder (w) visit(v) • In a postorder traversal, a node is visited after its descendants • Application: compute space used by files in a directory and its subdirectories 9 cs16/ 8 3 7 todo.txt1K homeworks/ programs/ 4 5 6 1 2 Robot.java20K h1c.doc3K h1nc.doc2K DDR.java10K Stocks.java25K

22. Inorder Traversal • For binary tree, inorder traversal can also be defined: L V R.

23. Java Implementation public static void inorder(TreeNode root) { if (root != null) { inorder(root.getChildAt(0)); visit(root); // assume declared. inorder(root.getChildAt(1)); }

24. Expression Notation • There are three common notations for expressions: • Prefix: operator come before operands. • Postfix: operator come after operands. • Infix: • Binary operator come between operands. • Unary operator come before the operand.

25. Examples • Infix: (a+b)*(c-d/f/(g+h)) • Used by human being. • Prefix: *+ab-c//df+gh • E.g. functional language such as Lisp: • Lisp: (* (+ a b) (- c (/ d (/ f (+ g h))))) • Postfix: ab+cdf/gh+/-* • E.g. Postfix calculator

26. Expression Trees • An expression tree can be used to represent an expression. • Operator: root • Operands: child sub-trees • Variable and constants: leaf nodes.

27. Traversals of Expression Trees • Preorder traversal => prefix notation. • Postorder traversal => postfix notation. • Inorder traversal with parenthesis added => infix notation.

28.  Linked Structure for Trees • A node is represented by an object storing • Element • Parent node • Sequence of children nodes • Node objects implement the Position ADT B   A D F B A D F   C E C E

29. Tree Classes in Ruby • FXTreeItem: a tree node for GUI application. • Some methods: • Tree related: childOf?, parentOf?, numChildren, parent, etc. • Traversal of child nodes: first, next, last, etc. • GUI related: selected?, selected=, hasFocus?, etc.

30. Tree Classes in Ruby 2 • FXTreeList: a list of FXTreeItem. • Include methods for list manipulation, such as appendItem, clearItems, etc.

31. Questions and Comments?