1. Trees by Dr. Bun Yue Professor of Computer Science yue@uhcl.edu http://sce.uhcl.edu/yue/ 2013 yue@uhcl.edu http://sce.uhcl.edu/yue/ CSCI 3333 Data Structures

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

3. 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 Computers”R”Us SalesR&DManufacturing LaptopsDesktops US International EuropeAsiaCanada

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. 5 British Royal Family Tree Trees Queen Victoria King Edward VII Albert Victor Alice King George V King George VIKing Edward VIII Queen Elizabeth II CharlesAnnAndrewEdward WilliamHenryPeterZara BeatriceEugenie

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. 8 Are these trees? Trees 12 3 4

9. subtree Tree Terminology  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. A B DC GH E F IJ K  Subtree: tree consisting of a node and its descendants

10. Tree ADT (§ 6.1.2)  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)  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

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  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 Make Money Fast! 1. MotivationsReferences2. Methods 2.1 Stock Fraud 2.2 Ponzi Scheme 1.1 Greed1.2 Avidity 2.3 Bank Robbery 1 2 3 5 4 678 9 Algorithm preOrder(v) visit(v) for each child w of v preorder (w)

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  In a postorder traversal, a node is visited after its descendants  Application: compute space used by files in a directory and its subdirectories Algorithm postOrder(v) for each child w of v postOrder (w) visit(v) cs16/ homeworks/ todo.txt 1K programs/ DDR.java 10K Stocks.java 25K h1c.doc 3K h1nc.doc 2K Robot.java 20K 9 3 1 7 2 456 8

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 D A CE F B  ADF  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?