1. Binary TreesCS-2303, C-Term 20101 Binary Trees (and Big “O” notation) CS-2303 System Programming Concepts (Slides include materials from The C Programming Language, 2 nd edition, by Kernighan and Ritchie and from C: How to Program, 5 th and 6 th editions, by Deitel and Deitel)

2. Binary TreesCS-2303, C-Term 20102 Definitions Linked List A data structure in which each element is dynamically allocated and in which elements point to each other to define a linear relationship Singly- or doubly-linked Stack, queue, circular list Tree A data structure in which each element is dynamically allocated and in which each element has more than one potential successor Defines a partial order

3. Binary TreesCS-2303, C-Term 20103 Binary Tree A linked list but with two links per item struct treeItem { type payload; treeItem *left; treeItem *right; }; leftright payload leftright payload leftright payload leftright payload leftright payload leftright payload leftright payload

4. Binary TreesCS-2303, C-Term 20104 Binary Tree (continued) Binary tree needs a root struct treeItem { type payload; treeItem *left; treeItem *right; }; struct treeItem *root; Binary trees often drawn with root at top! Unlike ordinary trees in the forest More like the root systems of a tree

5. Binary TreesCS-2303, C-Term 20105 Definitions (continued) Binary Tree A tree in which each element has two potential successors Subtree The set of nodes that are successors to a specific node, either directly or indirectly Root of a tree The node of the tree that is not the successor to any other node, all other nodes are (directly or indirectly) successors to it See Deitel & Deitel, §12.7 K & R, §6.5

6. Binary TreesCS-2303, C-Term 20106 Binary Tree struct treeItem { type payload; treeItem *left; treeItem *right; }; struct treeItem *root; leftright payload leftright payload leftright payload leftright payload leftright payload leftright payload leftright payload

7. Binary TreesCS-2303, C-Term 20107 Purpose of a Tree (Potentially) a very large data structure Capable of storing very many items In an orderly way Need to find items by value I.e., need to search through the data structure to see if it contains an item with the value we want Need to add new items If value is not already in the tree, add a new item … …so that it can be easily found in future Why not use a linked list?

8. Binary TreesCS-2303, C-Term 20108 Searching and Adding to a Binary Tree Look recursively down sequence of branches until either –Desired node is found; or –Null branch is encountered Replace with ptr to new item Decide which branch to follow based on payload leftright payload leftright payload leftright payload leftright payload leftright payload leftright payload leftright payload

9. Binary TreesCS-2303, C-Term 20109 Example — Searching a Tree typedef struct _treeItem { char *word;// part of payload int count;// part of payload _treeItem *left, *right; } treeItem; treeItem *findItem(treeItem *p, char *w) { if (p == NULL) return NULL;// item not found int c = strcmp(w, p->word); if (c == 0) return p; else if (c left, w); else return findItem(p->right, w); }

10. Binary TreesCS-2303, C-Term 201010 Why do this? Example — Adding an Item treeItem *addItem(treeItem *p, char *w) { if (p == NULL){ p = malloc(sizeof(treeItem)); char *c = malloc(strlen(w)+1); p->word = strcpy(c, w); p->count = 1; p->left = p->right = NULL; return p; }; int c = strcmp(w, p->word); if (c == 0) p->count++; else if (c left = addItem(p->left, w); else p->right = addItem(p->right, w); return p; }

11. Binary TreesCS-2303, C-Term 201011 Binary Tree Question:– how many calls to addItem for a tree with 10 6 nodes? –Assume balanced –I.e., approx same number of nodes on each subtree leftright payload leftright payload leftright payload leftright payload leftright payload leftright payload leftright payload

12. Binary TreesCS-2303, C-Term 201012 Answer Approximately 20 calls to addItem Note:– –2 10 = 1024  10 3 –Therefore 10 6  2 20 –Therefore it takes approximately 20 two-way branches to cover 10 6 items! How many comparisons would it take to search a linked list of 10 6 items?

13. Binary TreesCS-2303, C-Term 201013 Observation Problems like this occur in real life all the time Need to maintain a lot of data Usually random Need to search through it quickly Need to add (or delete) items dynamically Need to sort “on the fly” I.e., as you are adding and/or deleting items

14. Binary TreesCS-2303, C-Term 201014 Questions?

15. Binary TreesCS-2303, C-Term 201015 Binary Trees (continued) Binary tree does not need to be “balanced” i.e., with approximate same # of nodes hanging from right or left However, it often helps with performance Multiply-branched trees Like binary trees, but with more than two links per node

16. Binary TreesCS-2303, C-Term 201016 Binary Trees (continued) Binary tree does not need to be “balanced” i.e., with approximate same # of nodes hanging from right or left However, it helps with performance Time to reach a leaf node is O(log 2 n), where n is number of nodes in tree Multiply-branched trees Like binary trees, but with more than two links per node “Big-O” notation:– means “order of”

17. Binary TreesCS-2303, C-Term 201017 Order of Traversing Binary Trees In-order Traverse left sub-tree (in-order) Visit node itself Traverse right sub-tree (in-order) Pre-order Visit node first Traverse left sub-tree Traverse right sub-tree Post-order Traverse left sub-tree Traverse right sub-tree Visit node last

18. Binary TreesCS-2303, C-Term 201018 Question Suppose we wish to print out the strings stored in the tree of the previous example in alphabetical order? What traversal order of the tree should we use?

19. Binary TreesCS-2303, C-Term 201019 Another Example of Binary Tree x = (a.real*b.imag - b.real*a.imag) / sqrt(a.real*b.real – a.imag*b.imag) = x/ sqrt - **.. arealbimag.. brealaimag - …

20. Binary TreesCS-2303, C-Term 201020 Question What kind of traversal order is required for this expression? In-order? Pre-order? Post-order?

21. Binary TreesCS-2303, C-Term 201021 Binary Trees in Compilers Used to represent the structure of the compiled program Optimizations Common sub-expression detection Code simplification Loop unrolling Parallelization Reductions in strength – e.g., substituting additions for multiplications, etc. Many others Note: Deitel & Deitel, Ch 12 exercises, contain a series on building a compiler

22. Binary TreesCS-2303, C-Term 201022 Questions?

23. Binary TreesCS-2303, C-Term 201023 “Big O” notation New Topic

24. Binary TreesCS-2303, C-Term 201024 Linked Lists Again Linear data structure Easy to grow and shrink Easy to add and delete items Time to search for an item – O(n) I.e., proportional to n, the number of items in the list

25. Binary TreesCS-2303, C-Term 201025 Binary Trees Again Non-linear data structure Easy to grow and shrink Easy to add and delete items Time to search for an item – O(log n) I.e., proportional to log of number of items in the list

26. Binary TreesCS-2303, C-Term 201026 Definition — Big-O “Of the order of …” A characterization of the number of operations in an algorithm in terms of a mathematical function of the number of data items involved O(n) means that the number of operations to complete the algorithm is proportional to n E.g., searching a list with n items requires, on average, n/2 comparisons with payloads

27. Binary TreesCS-2303, C-Term 201027 Big-O (continued) O(n): proportional to n – i.e., linear O(n 2 ): proportional to n 2 – i.e., quadratic O(k n ) – proportional to k n – i.e., exponential … O(log n) – proportional to log n – i.e., sublinear O(n log n) Worse than O(n), better than O(n 2 ) O(1) – independent of n; i.e., constant

28. Binary TreesCS-2303, C-Term 201028 Anecdote & Questions:– In the design of electronic adders, what is the order of the carry-propagation? What is the order of floating point divide? What is the order of floating point square root? What program have we studied in this course that is O(2 n )? i.e., exponential?

29. Binary TreesCS-2303, C-Term 201029 Questions on Big-O?