Concept of Trees

1. Data Structure: Chapter 6 Min Chen School of Computer Science and Engineering Seoul National University Concept of Trees

2. Content • Definition of Trees • Representing Rooted Tree • Tree Traversal • Preorder Traversal • Postorder Traversal • Level Order Traversal

3. Definition of Trees • Tree: • Set of nodes and edges that connect them • Exactly one path between any 2 nodes • Rooted Tree: • One distinguished node is called the root • Every node C, except root, has one parent P, the first node on path from c to the root. C is P’s child • Root has no parent • A node can have any number of children

4. Some Definitions • Leaf • Node with no children • Siblings • Nodes with same parent • Ancestors • nodes on path from d to rott, including d, d’s parent, d’s grand parent, … root • Descendant • If A is B’s ancestor, then B is A’s Descendant

5. Some Definitions (2) • Length of path • Number of edges in path • Depth of node n • Length of path from n to root • Depth of root is zero • Height of node n • Length of path from n to its deepest descendant • Height of any leaf is zero • Height of a tree= Height of the root • Subtree rooted at n • The tree formed by n and its descendants

6. Representing Rooted Trees • G & T • Each node has 3 references stored in a list • Item • Parent • Children • Another Option: Sibling Tree • Siblings are directly linked

7. Sibling Tree Parent Class SibTreeNode { Object item ; SibTreeNode parent ; SibTreeNodefirstChild; SibTreeNodenextSibling; } Item First Child Next Sibling

8. Sibling Tree Parent Parent Parent Parent Parent Parent Parent Parent Item Item Item Item Item Item Item Item First Child First Child First Child First Child First Child First Child First Child First Child Next Sibling Next Sibling Next Sibling Next Sibling Next Sibling Next Sibling Next Sibling Next Sibling

9. Tree Traversal • Rooted Tree • Preorder Traversal • Postorder Traversal • Level Order Traversal • Binary Tree • InorderTraveral

10. Preorder Traversal • Visit each node before recursively visiting its children, left to right B has no child, no sibling A A A has no child, then sibling A’s First Child C B B C C’s First Child B’s First Child D G D E F G H G has no child, then sibling D has no child, then sibling E H H has no child, no sibling E has no child, then sibling F F has no child, no sibling

11. Preorder Traversal • Preorder Traversal Realization by Recursion Class SibTreeNode { public void preorder() { this.visit(); if(firstChild!=null){ firstChild.preorder(); } if(nextSibling!=null){ nextSibling.preorder(); } } }

12. Preorder Traversal • Preorder Traversal Realization by Stacks Stack: A B C D E G B F D E F G H C H A Visiting Sequence: A B D E F C G H

13. Postorder Traversal • Visit each node’s children (left-to-right) before the node itself A B C D E F G H

14. Postorder Traversal • Postorder Traversal Realization by Recursion Class SibTreeNode { public void postorder() { if(firstChild!=null) { firstChild.postorder(); } this.visit(); if(nextSibling!=null) { nextSibling.postorder(); } } }

15. Level Order Traversal • Visit root, then all (height-1) nodes, then all (height-2) nodes, … etc. A B C D E F G H

16. Level Order Traversal • Level Order Traversal Realization by Queues Queue: A H B C G F D E F G H E D C B A Visiting Sequence: A B C D E F G H

17. Binary Tree • A Binary Tree • No node has > 2 children • Every child is either left child or a right child, even if it is the only child

18. Representing Binary Tree • Binary Tree Node Parent Item Class BiTreeNode { Object item ; BiTreeNode parent ; BiTreeNodeleftChild; BiTreeNoderightChild; } Left Child Right Child

19. A Binary Tree Parent Parent Parent Parent Parent Parent Item Item Item Item Item Item Left Child Left Child Left Child Left Child Left Child Left Child Right Child Right Child Right Child Right Child Right Child Right Child

20. Inorder Traversal for Binary Tree • Visit left child, then node, then right child Class BiTreeNode{ public void inorder() { if(leftChild!=null){ leftChild.inorder(); } this.visit(); if(rightChild!=null) { rightChild.inorder(); } } }

21. Inorder Traversal for Binary Tree • Visualization of inorder traversal A B C D E F

22. Thank you!