Understanding Trees in Computer Science: Exploring Sort Algorithms and Tree Structures

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

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

3. 0 1 2 3 4 5 6 7 C O M P U T E R 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 ) } Real Java Code, Textbook, p. 394-396

4. 4 2 2 2 1 1 1 1 1 1 Revisision - Merge Sort 8 items 4 2 1 1 Analysis, Textbook, p. 393-398

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 0 1 2 3 4 8 3 9 1 7 3 1 7 8 9 Textbook, p. 399

7. Quicksort ≥p p <p Textbook, p. 399

8. 0 1 2 3 4 5 6 7 C O M P U T E R Java Code, Textbook, pp. 405-407, Description, Textbook, pp. 398-400

9. 0 1 2 3 4 5 6 7 C O M P U T E R 0 1 2 3 4 5 6 7 C O M P U T E R Java Code, Textbook, pp. 405-407, Description, Textbook, pp. 398-400

10. 0 1 2 3 4 5 6 7 C O M P U T E R 0 1 2 3 4 5 6 7 C O M P U T E R 1 2 3 4 5 6 7 E M O U T E R Java Code, Textbook, pp. 405-407, Description, Textbook, pp. 398-400

11. 0 1 2 3 4 5 6 7 C O M P U T E R 0 1 2 3 4 5 6 7 C O M P U T E R 1 2 3 4 5 6 7 E M O U T E R 1 2 4 5 6 7 E M T E R U Java Code, Textbook, pp. 405-407, Description, Textbook, pp. 398-400

12. 0 1 2 3 4 5 6 7 C O M P U T E R 0 1 2 3 4 5 6 7 C O M P U T E R 1 2 3 4 5 6 7 E M O U T E R 1 2 4 5 6 7 E M T E R U 4 5 6 2 T E R M Java Code, Textbook, pp. 405-407, Description, Textbook, pp. 398-400

13. 0 1 2 3 4 5 6 7 C O M P U T E R 0 1 2 3 4 5 6 7 C O M P U T E R 1 2 3 4 5 6 7 E M O U T P R 1 2 4 5 6 7 E M T P R U 4 5 6 2 T P R M 4 5 P T Java Code, Textbook, pp. 405-407, Description, Textbook, pp. 398-400

14. Quicksort 0 1 2 3 4 5 6 7 C O M P U T E R C O M P U T E R C E M O U T P R C E M O T P R U C E M O T P T R C E M O P T U R C E M O P T U R Textbook, pp. 398-400

15. Quicksort Complexity 0 1 2 3 4 5 6 7 A B C D E F G H Textbook, pp. 408-410

16. Partitioning O M P U T E R Textbook, p. 401-404

17. Partitioning O M P U T E R O M P U T E R Textbook, p. 401-404

18. Partitioning O M P U T E R O M P U T E R Textbook, p. 401-404

19. Partitioning O M P U T E R O M P U T E R O M P U T E R Textbook, p. 401-404

20. Partitioning O M P U T E R O M P U T E R O M P U T E R O M P U T E R Textbook, p. 401-404

21. Partitioning O M P U T E R O M P U T E R O M P U T E R O M P U T E R O M E U T P R Textbook, p. 401-404

22. Partitioning O M P U T E R O M P U T E R O M P U T E R O M P U T E R O M E U T P R O M E U T P R Textbook, p. 401-404

23. Partitioning O M P U T E R O M P U T E R O M P U T E R O M P U T E R O M E U T P R O M E U T P R E M O U T P R Textbook, p. 401-404

24. Mergesort: Efficiency: Quicksort vs. Mergesort Quicksort:

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

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

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

28. A Tree

29. Nodes Edges Root Leaf Parent Child Siblings Anscestor Descendant Subtrees Height Terminology A B C D E F G H I Textbook, p. 423

30. A B C D E F G H I Recurisve Definition A tree is a root node attached to a set of trees

31. A B C D E F G H I Node: Subtree References public class TreeNode { Object item; TreeNode[] subTrees; }

32. Node: First Child Next Sibling public class TreeNode { Object item; TreeNode firstChild; TreeNode nextSibling; } A B C D E F G H I

33. Binary Trees Textbook, p. 423-4

34. Binary Trees 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 Textbook, p. 423-4

35. Binary Trees 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 Textbook, p. 423-4

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 A C Alternative definition, Textbook, p. 430-431

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  }