1. 12.Binary Search Trees Hsu, Lih-Hsing

2. Computer Theory Lab. Chapter 12P.2 12.1 What is a binary search tree? Binary-search property: Let x be a node in a binary search tree. If y is a node in the left subtree of x, then key[y]  key[x]. If y is a node in the right subtree of x, then key[x]  key[y].

3. Computer Theory Lab. Chapter 12P.3 Binary search Tree

4. Computer Theory Lab. Chapter 12P.4 Inorder tree walk INORDER_TREE_WALK(x) 1if 2then INORDER_TREE_WALK(left[x]) 3print key[x] 4INORDER_TREE_WALK(right[x])

5. Computer Theory Lab. Chapter 12P.5 Theorem 12.1 If x is the root of an n-node subtree, then the call INORDER-TREE-WALK(x) takes  (n) time.

6. Computer Theory Lab. Chapter 12P.6 Preorder tree walk Postorder tree walk

7. Computer Theory Lab. Chapter 12P.7 12.2 Querying a binary search tree

8. Computer Theory Lab. Chapter 12P.8 TREE_SEARCH(x,k) 1if or 2then return x 3if 4then return TREE_SEARCH(left[x],k) 5else return TREE_SEARCH(right[x],k)

9. Computer Theory Lab. Chapter 12P.9 ITERATIVE_SEARCH(x,k) 1While or 2do if 3then 4then 5return x

10. Computer Theory Lab. Chapter 12P.10 MAXIMUM and MINIMUM TREE_MINIMUM(x) 1 while left[x]  NIL 2 do x  left[x] 3 return x TREE_MAXIMUM(x) 1 while right[x]  NIL 2 do x  right[x] 3 return x

11. SUCCESSOR and PREDECESSOR

12. Computer Theory Lab. Chapter 12P.12 TREE_SUCCESSOR 1 if 2 then return TREE_MINIMUM(right[x]) 3 4 while and 5 do 6 7 return y

13. Computer Theory Lab. Chapter 12P.13 Theorem 12.2 The dynamic-set operations, SEARCH, MINIMUM, MAXIMUM, SUCCESSOR, and PREDECESSOR can be made to run in O(h) time on a binary search tree of height h.

14. 12.3 Insertion and deletion

15. Computer Theory Lab. Chapter 12P.15 Tree-Insert(T,z) 1 y  NIL 2 x  root[T] 3 while x  NIL 4 do y  x 5if key[z] < key[x] 6 then x  left[x] 7 else x  right[x] 8 p[z]  y Insertion

16. Computer Theory Lab. Chapter 12P.16 9 if y = NIL 10 then root[T]  z  tree T was empty 11 else if key[z] < key[y] 12 then left[y]  z 13else right[y]  z

17. Computer Theory Lab. Chapter 12P.17 Inserting an item with key 13 into a binary search tree

18. Computer Theory Lab. Chapter 12P.18 Tree-Delete(T,z) 1 if left[z] = NIL or right[z] = NIL 2 then y  z 3 else y  Tree-Successor(z) 4 if left[y]  NIL 5then x  left[y] 6else x  right[y] 7 if x  NIL 8 then p[x]  p[y] Deletion

19. Computer Theory Lab. Chapter 12P.19 9 if p[y] = NIL 10 then root[T]  x 11else if y = left[p[y]] 12then left[p[y]]  x 13else right[p[y]]  x 14 if y  z 15then key[z]  key[y] 16 copy y’s satellite data into z 17 return y

20. Computer Theory Lab. Chapter 12P.20 z has no children

21. Computer Theory Lab. Chapter 12P.21 z has only one child

22. Computer Theory Lab. Chapter 12P.22 z has two children

23. Computer Theory Lab. Chapter 12P.23 Theorem 12.3 The dynamic-set operations, INSERT and DELETE can be made to run in O(h) time on a binary search tree of height h.