1. Binary Search Tree C and Data Structures Baojian Hua bjhua@ustc.edu.cn

2. Dictionary-like Data Structure A dictionary-like data structure contains a collection of tuple data: data = key is comparable and distinct supports these operations: new () insert (dict, k, v) lookup (dict, k) delete (dict, k) We discussed a linear list-based representation of dictionary, this class studies another strategy based on binary trees

3. Binary Search Tree A binary search tree is a binary tree satisfies: every node contain data=, and every key is unique all keys in left sub-tree is less than that in the root all keys in right sub-tree is greater than that in the root both the left and right sub-trees are also binary search trees

4. Example 40 2060 10 3050 5 70 55

5. Operations All the tree operations we ’ ve discussed also apply to binary search tree And BST also supports (as a general dictionary-like data structure): search (bst, key) insert (bst, key, value) delete (bst, key)

6. Abstract Data Types in C: Interface // in file “bst.h” #ifndef BST_H #define BST_H #define T Bst_t typedef struct T *T; T Bst_new (); T Bst_insert (T t, poly key, poly value); poly Bst_lookup (T t, poly key); #undef T #endif

7. Implementation // in file “bst.c” #include “bst.h” #define T Bst_t struct T { poly data; T left; T right; }; t leftkeyrightv

8. Operations: “ new ” T Bst_new () { return 0; }

9. How to search in a BST?--- lookup (bst, key) 40 2060 10 3050 5 70 55

10. Operations: “ lookup ” poly Bst_lookup (T t, poly key) { if (0==t) return 0; if (key == t->key) // what’s “==“ ? return t->value; if (key key) // what’s “<“? return lookup (t->left, key); return lookup (t->right, key); }

11. Example: search 55 40 2060 10 3050 5 70 55

12. Example: search 55 40 2060 10 3050 5 70 55

13. Example: search 55 40 2060 10 3050 5 70 55

14. Example: search 55 40 2060 10 3050 5 70 55

15. Example: search 55 40 2060 10 3050 5 70 55

16. Example: search 55 40 2060 10 3050 5 70 55

17. Example: search 55 40 2060 10 3050 5 70 55

18. Example: search 55 40 2060 10 3050 5 70 55

19. An Iterative Version poly lookup2 (T t, poly key) { if (0==t) return 0; T p = t; while (p && p->key!=key){ // what’s “!=“? if (key key) p = p->left; else p = p->right; } return p; }

20. Adding New Bindings: insert (bst t, poly key, poly value) Main idea: search the tree, if these already exists a key k ’ ==k, then insertion fails else if tree t==NULL, return a new bst else search a proper position to insert the tuple What ’ s a proper position?

21. Example: insert 45 40 2060 10 3050 5 70 5545

22. Example: insert 45 40 2060 10 3050 5 70 5545 searchParent

23. A Functional Version T insert (T t, poly key, poly value) { if (!t) return Bst_new2 (0, 0, key, value); switch (compare (key key)) { case “==“: error (“key already exists”); case “<“: return Bst_new2 (insert (t->left, key, value), t->right, t->key, t->value); case “>”: return Bst_new2 (t->left, insert (t->right, key,value), t->key, t->value); }

24. Example: insert 45 40 2060 10 3050 5 70 5545 50 60 40

25. remove case#1: leaf node 40 2060 10 3050 5 70 55

26. remove case#2: 1-degree node 40 2060 10 3050 5 70 55

27. remove case#3: 2-degree node 40 2060 10 3050 5 70 55

28. remove case#3: 2-degree node 40 2055 10 3050 5 70 60