1 Lecture 19: Trees Lecturer: Santokh Singh CompSci 105 SS 2005 Principles of Computer Science

2 Revision - Sorting O(n)O(n) Faster Code O(n2)O(n2) O(n3)O(n3) O(n log n) O(log n) O(1) O(2n)O(2n) Merge Sort Selection Sort

3 Real Java Code, Textbook, p mergeSort( theArray, first, last ) if (first < last ) { mid = (first + last ) / 2 mergesort(theArray, first, mid) mergesort(theArray, mid+1, last) merge(theArray(first, mid, last ) } MOCP 0123 ETUR 4567

4 Revisision - Merge Sort Analysis, Textbook, p items

5 Quick Sort Algorithm Analysis Trees Introduction General Tree Structures Binary Trees Reference-Based Implementation

6 Revision - Partitioning (as seen in L8) <p p ≥p≥p Textbook, p. 399

7 Quicksort <p p ≥p≥p Textbook, p. 399

8 MOCP 0123 ETUR 4567 Java Code, Textbook, pp , Description, Textbook, pp

9 MOCP 0123 ETUR 4567 MOCP 0123 ETUR 4567

10 MOCP 0123 ETUR 4567 Java Code, Textbook, pp , Description, Textbook, pp MOCP 0123 ETUR 4567 MEO 123 ETUR 4567

11 MOCP 0123 ETUR 4567 Java Code, Textbook, pp , Description, Textbook, pp MOCP 0123 ETUR 4567 MEO 123 ETUR 4567 ME 12 RETU 4567

12 MOCP 0123 ETUR 4567 Java Code, Textbook, pp , Description, Textbook, pp MOCP 0123 ETUR 4567 MEO 123 ETUR 4567 ME 12 M 2 RETU 4567 RET 456

13 MOCP 0123 ETUR 4567 Java Code, Textbook, pp , Description, Textbook, pp MOCP 0123 ETUR 4567 MEO 123 PTUR 4567 ME 12 M 2 RPTU 4567 RPT 456 TP 45

14 Quicksort MOCP 0123 ETUR 4567 MOCPETUR MECOPTUR MECORPTU MECOTPTR Textbook, pp MECOUTPR MECOUTPR

15 Quicksort Complexity CBAD 0123 GFEH 4567 Textbook, pp

16 Partitioning Textbook, p MOPETUR

17 Partitioning Textbook, p MOPETUR MOPETUR

18 Partitioning Textbook, p MOPETUR MOPETUR

19 Partitioning Textbook, p MOPETUR MOPETUR MOPETUR

20 Partitioning Textbook, p MOPETUR MOPETUR MOPETUR MOPETUR

21 Partitioning Textbook, p MOPETUR MOPETUR MOPETUR MOPETUR MOEPTUR

22 Partitioning Textbook, p MOPETUR MOPETUR MOPETUR MOPETUR MOEPTUR MOEPTUR

23 Partitioning Textbook, p MOPETUR MOPETUR MOPETUR MOPETUR MOEPTUR MOEPTUR MEOPTUR

24 Mergesort: Quicksort: Efficiency: Quicksort vs. Mergesort

25 Sorting O(n2)O(n2) O(n log n) Merge Sort Selection Sort Quicksort (Worst) Quicksort (Average)

26 Comparing Sorting Algorithms Merge Sort Bubble Sort Quicksort Selection Sort Insertion Sort

27 Quick Sort Algorithm Analysis Trees Introduction General Tree Structures Binary Trees Reference-Based Implementation

28 A Tree

29 Terminology A BDC EGFIH Textbook, p. 423 Nodes Edges Root Leaf Parent Child Siblings Anscestor Descendant Subtrees Height

30 Recurisve Definition A BDC EGFIH A tree is a root node attached to a set of trees

31 Node: Subtree References public class TreeNode { Object item; TreeNode[] subTrees; } A BDC EGFIH

32 Node: First Child Next Sibling public class TreeNode { Object item; TreeNode firstChild; TreeNode nextSibling; } A BDC EGFIH

33 Binary Trees Textbook, p

34 Binary Trees Textbook, p A binary tree is either empty or is a root node storing an item attached to a binary tree called the left subtree and a binary tree called the right subtree

35 Binary Trees Textbook, p A binary tree is either empty or is a root node storing an item attached to a binary tree called the left subtree and a binary tree called the right subtree

36 Binary Tree Node (Ref based) public class TreeNode { Object item; TreeNode left; TreeNode right; }

37 Binary Tree ADT TreeNode createBinaryTree( ) Object getRootItem( ) TreeNode getLeft ( ) TreeNode getRight ( ) setLeft ( TreeNode ) setRight ( TreeNode ) setRootItem( Object ) B AC Alternative definition, Textbook, p

38 // Example of painting beautiful binary trees in java applications:- public void paint(Graphics g){ if(root!= null) draw(1, getWidth()/2, 40,180,80,root, g ); // Recursive method } public void draw(int order, int x, int y, int xGap, int yGap,BinaryTreeNode e,Graphics g){ if (e.left()!=null){ int leftX = x-xGap; // draws to left now int leftY = // How do we draw child downwards in the application? g.drawLine(x,y,leftX,leftY); // draw the connecting line draw( order+1,leftX, leftY, xGap/2, yGap,e.left(),g); // recursion // int order need not be used – but can be used for depth } if (e.right()!=null){ // just do similarly for right child now } g.setColor(Color…..); // What circle border color do you like? g.fillOval(x-size, y-size, 2*size, 2*size); g.setColor(Color…..); // Inner color of circle g.fillOval(x-size+1, y-size+1, 2*size-2, 2*size-2); g.setColor(Color….); // Color of values displayed g.drawString(""+e.value(),…, …); // display the value correctly }