Binary Trees (and Big “O” notation)

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

2. 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 Binary Trees

3. A linked list but with two links per item struct treeItem {type payload;treeItem *left; treeItem *right; }; payload payload payload payload payload payload payload left left left left left left left right right right right right right right Binary Tree Binary Trees

4. 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 Binary Trees

5. Definitions (continued) See Deitel & Deitel, §12.7K & R, §6.5 • 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 Binary Trees

6. struct treeItem {type payload;treeItem *left; treeItem *right; }; struct treeItem *root; payload payload payload payload payload payload payload left left left left left left left right right right right right right right Binary Tree Binary Trees

7. 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? Binary Trees

8. 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 payload payload payload payload payload payload payload left left left left left left left right right right right right right right Searching and Adding to a Binary Tree Binary Trees

9. Example — Searching a Tree typedef struct _treeItem {char *word; // part of payloadint count; // part of payload_treeItem *left, *right;} treeItem; treeItem *findItem(treeItem *p, char *w) {if (p == NULL) return NULL; // item not foundint c = strcmp(w, p->word);if (c == 0) return p;else if (c < 0) return findItem(p->left, w);else return findItem(p->right, w); } Binary Trees

10. 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 < 0) p->left = addItem(p->left, w);else p->right = addItem(p->right, w);return p; } Why do this? Binary Trees

11. Question:– how many calls to addItem for a tree with 106 nodes? Assume balanced I.e., approx same number of nodes on each subtree payload payload payload payload payload payload payload left left left left left left left right right right right right right right Binary Tree Binary Trees

12. Answer • Approximately 20 calls to addItem • Note:– • 210 = 1024  103 • Therefore 106  220 • Therefore it takes approximately 20 two-way branches to cover 106 items! • How many comparisons would it take to search a linked list of 106 items? Binary Trees

13. 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 Binary Trees

14. Questions? Binary Trees

15. 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 Binary Trees

16. “Big-O” notation:– means “order of” 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(log2 n), where n is number of nodes in tree • Multiply-branched trees • Like binary trees, but with more than two links per node Binary Trees

17. 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 Binary Trees

18. 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? Binary Trees

19. 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 - - * * … . . . . a real b imag b real a imag Binary Trees

20. Question • What kind of traversal order is required for this expression? • In-order? • Pre-order? • Post-order? Binary Trees

21. 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 Binary Trees

22. Questions? Binary Trees

23. “Big O” notation New Topic Binary Trees

24. I.e., proportional to n, the number of items in the list 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) Binary Trees

25. I.e., proportional to log of number of items in the list 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) Binary Trees

26. 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 Binary Trees

27. Big-O(continued) • O(n): proportional to n – i.e., linear • O(n2): proportional to n2 – i.e., quadratic • O(kn) – proportional to kn – i.e., exponential • … • O(log n) – proportional to log n – i.e., sublinear • O(n log n) • Worse than O(n), better than O(n2) • O(1) – independent of n; i.e., constant Binary Trees

28. 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(2n)? i.e., exponential? Binary Trees

29. Questions on Big-O? Binary Trees