1. Data Structures and Algorithms for Information Processing
Lecture 5: Trees
Lecture 5: Trees

2. Binary Trees
A binary tree has nodes, similar to nodes in a linked list structure.
Data of one sort or another may be stored at each node.
Each node is either a leaf, having no children, or an internal node, with one or two children
In many ways, a tree is like the other structures you have seen: A tree consists of nodes, and each node can contain data of one sort or another.
Lecture 5: Trees

3. Binary Trees
Lecture 5: Trees

4. Some Terminology Root is the unique node with no parent
Leaves or terminals are nodes with no children
A subtree is a node together with all its descendents
Level of a node is number of nodes on path from the node to root
Lecture 5: Trees

5. Arithmetic Expressions A*(((B+C)*(D*E))+F)
Example use of Trees
Arithmetic Expressions
A*(((B+C)*(D*E))+F)
Each internal node labeled by an operator
Each leaf labeled by a variable or numeric value
Tree determines a unique value
Lecture 5: Trees

6. Example Use of Trees Representing Chinese Characters
Characters composed hierarchically into boxes
Boxes can be oriented left-right, top-down, or outside-inside
Each leaf labeled by one of radicals
Lecture 5: Trees

7. Binary vs. Binary-Unary
Important distinction : Sometimes binary trees are defined to allow internal nodes with one or two children.
There is a big difference...
Lecture 5: Trees

8. Counting Trees The number of binary trees with internal nodes is
There is no such formula if we allow unary internal nodes!
Lecture 5: Trees

9. Exercise Write efficient Java functions int numBTrees(int n)
int numBUTrees(int n)
that return the number of binary trees and binary/unary trees, respectively.
Lecture 5: Trees

10. Properties of Trees There is exactly one path connecting any two nodes
Any two nodes have a least common ancestor
A tree with N internal nodes has N+1 external nodes
(easy to prove by induction)
Lecture 5: Trees

11. The Best Trees
A binary tree is full if internal nodes fill every level, except possibly the last.
The height of a full binary tree with internal nodes is
Lecture 5: Trees

12. Complete Binary Trees A complete tree is one having levels and leaves
Complete trees have a simple implementation using arrays
(How?)
Lecture 5: Trees

13. Class for Binary Nodes public class BTNode { private Object data;
private BTNode left;
private BTNode right;
...
Lecture 5: Trees

14. Class for Binary Nodes public BTNode(Object obj, BTNode l, BTNode r) {
data = obj; left = l; right= r; }
public boolean isLeaf()
return (left == null) &&
(right == null);
...
Lecture 5: Trees

15. Copying Trees public static BTNode treeCopy(BTNode t) {
if (t == null) return null;
else
BTNode leftCopy =
treeCopy(t.left);
BTNode rightCopy =
treeCopy(t.right);
return new BTNode(t.data,
leftCopy,
rightCopy); }
Lecture 5: Trees

16. Tree Traversals Preorder traversal P M L E S R A A T E E
Lecture 5: Trees

17. Tree Traversals Inorder traversal P M L E S R A A T E E
Lecture 5: Trees

18. Tree Traversals Postorder traversal P M L E S R A A T E E
Lecture 5: Trees

19. Tree Traversals Level order traversal P M L E S R A A T E E
Lecture 5: Trees

20. Preorder Traversal Process the root
Process the nodes in the left subtree
Process the nodes in the right subtree
Lecture 5: Trees

21. Preorder Print public void preorderPrint() { System.out.println(data);
if (left != null)
left.preorderPrint();
if (right != null)
right.preorderPrint(); }
Lecture 5: Trees

22. Inorder Traversal Profess the nodes in the left subtree
Process the root
Process the nodes in the right subtree
Lecture 5: Trees

23. Inorder Print public void preorderPrint() { if (left != null)
left.preorderPrint();
System.out.println(data);
if (right != null)
right.preorderPrint(); }
Lecture 5: Trees