CISC220 Fall 2009 James Atlas

1. CISC220Fall 2009James Atlas Lecture 13: Trees

2. Skip Lists

3. Project 1 • AI • Graphics • Networking • Bio-informatics • Natural Language Processing

4. Objectives for Today • Understand Trees/Terminology • Use basic traversals on trees • Understand binary search trees • Construct and use binary search trees • Reading - K+W Chap 8

5. Trees • Nonlinear data structure

6. Tree Terminology • root, leaf • parent, child, sibling • subtree • external, internal node • ancestor, descendant • depth, height

7. Binary Trees • Each node has 0, 1, or 2 children

8. Binary Tree Application: Huffman Tree

9. Tree Traversal • process of visiting each node • 4 different standard traversals: • preorder • inorder • postorder • level-order (also called breadth-first) • Interactive example: • http://nova.umuc.edu/~jarc/idsv/lesson1.html

10. Traversal Exercise Find the: • preorder • inorder • postorder • level-order

11. Exercise Answers • Preorder traversal sequence: F, B, A, D, C, E, G, I, H (root, left, right) • Inorder traversal sequence: A, B, C, D, E, F, G, H, I (left, root, right) • Postorder traversal sequence: A, C, E, D, B, H, I, G, F (left, right, root) • Level-order traversal sequence: F, B, G, A, D, I, C, E, H

12. Binary Trees • Each node has 0, 1, or 2 children

13. Binary Search Trees Binary search trees • Elements in left subtree < element in subtree root • Elements in right subtree > element in subtree root • Both the left and right subtrees are binary search trees

14. Binary Search Tree (Example) • Is this a binary search tree?

15. Binary Search Tree (Example)

16. Searching a BST Search algorithm: • if the tree is empty, return NULL • if target equals to root node, return that data • if target < root node, return search(left subtree) • otherwise return search(right subtree)

17. find(7) find(12) find(0) find(14)

18. Inserting to a Binary Search Tree if root is NULL replace empty tree with new data leaf; else if item < root->data return insert(left subtree, item) else return insert(right subtree, item) • http://www.cs.jhu.edu/~goodrich/dsa/trees/btree.html

19. Inserting to a Binary Search Tree

20. Removing from a Binary Search Tree • Item not present: do nothing • Item present in leaf: remove leaf (change to null) • Item in non-leaf with one child: Replace current node with that child • Item in non-leaf with two children? • Find largest item in the left subtree • Recursively remove it • Use it as the parent of the two subtrees • (Could use smallest item in right subtree)

21. Removing from a Binary Search Tree

22. Order of operations?