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TREES, PART 2 Lecture 12 CS2110 Summer 2019 Announcements 2 The regrading period has opened for Assignment 1. Deadline: Saturday, July 13th at 5PM JavaHyperText topics 3 Tree traversals (preorder, inorder, postorder)

  1. TREES, PART 2 Lecture 12 CS2110 – Summer 2019

  2. Announcements 2 ¨ The regrading period has opened for Assignment 1. ¨ Deadline: Saturday, July 13th at 5PM

  3. JavaHyperText topics 3 ¨ Tree traversals (preorder, inorder, postorder) ¤ http://www.cs.cornell.edu/courses/JavaAndDS/files/tr ee6BTreeTraversal.pdf ¨ Stack machines ¤ http://www.cs.cornell.edu/courses/JavaAndDS/explain Java/03methodCalls.html

  4. Trees, re-implemented 4 ¨ Last time: lots of null comparisons to handle empty trees ¨ A more OO design: ¤ Interface to represent operations on trees ¤ Classes to represent behavior of empty vs. non-empty trees Demo

  5. Iterate through data structure 5 Iterate: process elements of data structure ¨ Sum all elements ¨ Print each element ¨ … Data Structure Order to iterate Array Forwards: 2, 1, 3, 0 2 1 3 0 Backwards: 0, 3, 1, 2 Linked List Forwards: 2, 1, 3, 0 2 1 3 0 Binary Tree ??? 2 1 3

  6. Iterate through data structure 6 5 2 8 0 3 7 9 Discuss: What would a reasonable order be?

  7. Tree Traversals 7

  8. Tree traversals 8 ¨ Iterating through tree is aka tree traversal ¨ Well-known recursive tree traversal algorithms: ¤ Preorder ¤ Inorder ¤ Postorder ¨ Another, non-recursive: level order (later in semester)

  9. Preorder “Pre:” process root before subtrees 1st value 2nd 3rd left right subtree subtree

  10. Inorder “In:” process root in-between subtrees 2nd value 1st 3rd left right subtree subtree

  11. Postorder “Post:” process root after subtrees 3rd value 2nd 1st left right subtree subtree

  12. Poll 12 Which traversal would print out this BST in ascending order? 5 2 8 0 3 7 9

  13. Example: Syntax Trees 13

  14. Syntax Trees 14 ¨ Trees can represent (Java) expressions ¨ Expression: 2 * 1 – (1 + 0) ¨ Tree: – * + 1 2 1 0

  15. Traversals of expression tree 15 – * + 2 1 1 0 Preorder traversal - * 2 1 + 1 0 1. Visit the root 2. Visit the left subtree 3. Visit the right subtree

  16. Traversals of expression tree 16 – * + 2 1 1 0 Preorder traversal - * 2 1 + 1 0 2 1 * 1 0 + - Postorder traversal 1. Visit the left subtree 2. Visit the right subtree 3. Visit the root

  17. Traversals of expression tree 17 – * + 2 1 1 0 Preorder traversal - * 2 1 + 1 0 2 1 * 1 0 + - Postorder traversal 2 * 1 - 1 + 0 Inorder traversal 1. Visit the left subtree 2. Visit the root 3. Visit the right subtree

  18. Traversals of expression tree 18 – * + 2 1 1 0 Preorder traversal - * 2 1 + 1 0 2 1 * 1 0 + - Postorder traversal 2 * 1 - 1 + 0 Inorder traversal Original expression, except for parens

  19. Prefix notation 19 ¨ Function calls in most programming languages use prefix notation: e.g., add(37, 5) . ¨ Aka Polish notation (PN) in honor of inventor, Polish logician Jan Łukasiewicz ¨ Some languages (Lisp, Scheme, Racket) use prefix notation for everything to make the syntax uniform. (- (* 2 1) (+ 1 0)) (define (fib n) (if (<= n 2) 1 (+ (fib (- n 1) (fib (- n 2)))))

  20. Postfix notation 20 ¨ Some languages (Forth, PostScript, HP calculators) use postfix notation ¨ Aka reverse Polish notation (RPN) 2 1 mul 1 0 add sub /fib { dup 3 lt { pop 1 } { dup 1 sub fib exch 2 sub fib add } ifelse } def

  21. Back to Trees 21

  22. Recover tree from traversal 22 Suppose inorder is B C A E D. Can we recover the tree uniquely? Discuss.

  23. Recover tree from traversal 23 Suppose inorder is B C A E D. Can we recover the tree uniquely? No! C A B B E E A D C D

  24. Recover tree from traversals 24 Suppose inorder is B C A E D preorder is A B C D E Can we determine the tree uniquely?

  25. Recover tree from traversals 25 Suppose inorder is B C A E D preorder is A B C D E Can we determine the tree uniquely? Yes! ¨ What is root? Preorder tells us: A ¨ What comes before/after root A? Inorder tells us: ¤ Before: B C ¤ After: E D ¨ Now recurse! Figure out left/right subtrees using same technique.

  26. Recover tree from traversals 26 Suppose inorder is B C A E D preorder is A B C D E How can we determine the tree uniquely? Discuss.

  27. Recover tree from traversals 27 Suppose inorder is B C A E D preorder is A B C D E Root is A; left subtree contains B C; right contains E D Left: Right: Inorder is B C Inorder is E D Preorder is B C Preorder is D E • • What is root? Preorder: B What is root? Preorder: D • • What is before/after B? Inorder: What is before/after D? Inorder: • • Before: nothing Before: E • • After: C After: nothing

  28. Recover tree from traversals 28 Suppose inorder is B C A E D preorder is A B C D E Tree is A B D C E

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