Binary Search Trees

1. Binary Search Trees

2. Definition Of Binary Search Tree A binary tree. Each node has a (key, value) pair. For every node x, all keys in the left subtree of x are smaller than that in x. For every node x, all keys in the right subtree of x are greater than that in x.

3. 20 Example Binary Search Tree 10 40 6 15 30 25 2 8 Only keys are shown.

4. ADT bsTree AbstractDataTypebsTree { Instances Operations Find(k) Insert(p) Erase(k) Ascend() // in-order tree traversal }

5. 20 10 40 6 15 30 25 2 8 The Operation ascend() Do an inorder traversal.

6. 20 10 40 6 15 30 25 2 8 The Operation find(k)

7. 20 10 40 6 15 30 25 2 8 The Operation insert() 35 Insert a pair whose key is 35.

8. 20 10 40 6 15 30 25 35 2 8 The Operation insert() 7 Insert a pair whose key is 7.

9. 20 The Operation insert() 10 40 6 15 30 18 25 35 2 8 7 Insert a pair whose key is 18.

10. 20 The Operation insert() 10 40 6 15 30 18 25 35 2 8 7 Complexity of insert() is O(height).

11. Three cases: • Element is in a leaf. • Element is in a degree 1 node (has one child). • Element is in a degree 2 node (has two children). The Operation erase()

12. 20 10 40 6 15 30 18 25 35 2 8 7 Erase From A Leaf Erase a leaf element. key = 7

13. 20 10 40 6 15 30 18 25 35 2 8 7 Erase From A Leaf (contd.) Erase a leaf element. key = 35

14. Remove node with one child • If node n has one child, move n’s child up to take n’s place.

15. 20 10 40 6 15 30 18 25 35 2 8 7 Erase From A Degree 1 Node Erase from a degree 1 node. key = 15

16. 20 10 40 6 18 30 25 35 2 8 7 Erase From A Degree 1 Node (contd.) Erase from a degree 1 node. key = 15

17. 20 10 40 6 15 30 18 25 35 2 8 7 Erase From A Degree 1 Node(contd.) Erase from a degree 1 node. key = 40

18. Remove node with two children • If node n has two children, let x be node in n’sright subtree with smallest key, • Remove x • Replace n’s key with x’s key 20 10 40

19. Remove node with two children • If node n has two children, let x be node in n’sleft subtree with largest key, • Remove x • Replace n’s key with x’s key 20 10 40

20. 20 10 40 6 15 30 18 25 35 2 8 7 Erase From A Degree 2 Node Erase from a degree 2 node. key = 10

21. 20 Erase From A Degree 2 Node 10 40 6 15 30 18 25 35 2 8 7 Replace with largest key in left subtree(or smallest in right subtree).

22. 20 Erase From A Degree 2 Node 10 40 6 15 30 18 25 35 2 8 7 Replace with largest key in left subtree (or smallest in right subtree).

23. 20 Erase From A Degree 2 Node 8 40 6 15 30 18 25 35 2 8 7 Replace with largest key in left subtree (or smallest in right subtree).

24. 20 Erase From A Degree 2 Node 8 40 6 15 30 18 25 35 2 8 7 Largest key must be in a leaf or degree 1 node.

25. 20 10 40 6 15 30 18 25 35 2 8 7 Another Erase From A Degree 2 Node Erase from a degree 2 node. key = 20

26. 20 Erase From A Degree 2 Node 10 40 6 15 30 18 25 35 2 8 7 Replace with largest in left subtree.

27. 20 Erase From A Degree 2 Node 10 40 6 15 30 18 25 35 2 8 7 Replace with largest in left subtree.

28. 18 Erase From A Degree 2 Node 10 40 6 15 30 18 25 35 2 8 7 Replace with largest in left subtree.

29. 18 Erase From A Degree 2 Node 10 40 6 15 30 25 35 2 8 7

30. Exercise • Start with an empty binary search tree. • Insert the keys 4,12,8,16,6,18,24,2,14,3, draw the tree following each insert. • From the tree above, delete the keys, 6,14,16,4 in order, draw the search tree following each deletion.

32. // abstract class bsTree // abstract data type specification for binary search trees // all methods are pure virtual functions // K is key type and E is value type #ifndefbsTree_ #define bsTree_ #include "dictionary.h" using namespace std; template<class K, class E> class bsTree : public dictionary<K,E> { public: virtual void ascend() = 0; // output in ascending order of key }; #endif