1. Trees

2. Tree Consists of a set of nodes and directed edges connecting them
Rooted trees
One node is distinguished as root
Every node c, except root, has exactly one incoming edge from another node p. p is c’s parent.
There exists a unique path from root to any node

3. Tree Properties and Definitions
A tree with N nodes has N-1 edges
Siblings – Nodes with same parent
Ancestor & descendant
If there is a path from u to v, v is the descendant of u & u is the ancestor of v
If u ≠ v, v is proper descendant of u & u is proper ancestor of v

4. Definitions (Contd.)
Depth of a node – Length of path from root to node
Depth of root is 0
Depth of any node is one more than its parent’s depth
Height of a node – Length of path from node to deepest descendant leaf
Size of a node – Number of descendants of the node (including itself)
Height of a tree – Height of the root of the tree

5. Example

6. Recursive Definition A tree is Either empty
Or it consists of root and zero or more of subtrees
Roots of subtrees connected by an edge to root of tree

7. Illustration

8. Implementing Trees Straightforward method
Every node in the tree has link to each of its children
Number of children can vary greatly
Child count may not be known before hand
Leads to space wastage
First Child – Next Sibling method
Children are stored as linked lists
Node stores link to leftmost child and right sibling

9. Illustration

10. Binary Tree A tree in which each node has at most two children
Left child and right child
Recursive definition
A binary tree is either empty, or it consists of a root, a left tree and a right tree
Applications
Expression trees in compilers
Huffman coding tree
Binary search trees

11. Applications

12. Binary Tree Implementation
Binary node
Stores an individual node in a binary tree
Consists of data (object to store), references to right child node and left child node (both binary nodes)
Binary tree
Stores the root node of the tree

13. BinaryNode and BinaryTree
public class BinaryNode{
private Object element;
private BinaryNode left;
private BinaryNode right; }
public class BinaryTree{
private BinaryNode root;

14. Methods in BinaryNode Two constructors
First one takes in element, right node, and left node
Second one does not take in any parameters (all are set to null)
Setters for element, left, and right
Getters for element, left, and right
Computing size and height of nodes
Tree traversal methods (Preorder, Inorder & Postorder)

15. Computing Size Recursive method that terminates at leaf nodes
public int size(){
if(left == null && right == null)
return(1);
if(left == null) return(right.size() + 1);
if(right == null) return(left.size() + 1);
return(left.size() + right.size() + 1); }

16. Computing Height Recursive method terminating in leaves
public int height(){
if(left == null && right == null) return(0);
if(left == null) return(right.height() + 1);
if(right == null) return(left.height() + 1);
if(right.height() => left.height()) return(right.height() + 1);
return(left.height() + 1);

17. Tree Traversals Accessing (or processing) individual nodes of a tree
Usually implemented recursively
Three kinds of tree traversal
Preorder
Inorder
Postorder
Our example considers printing node information

18. Preorder Traveral Process current node Process left subtree
Process right subtree
public void printPreOrder(){
System.out.println(element.toString());
if (left != null) left.printPreOrder();
if (right != null) right.printPreOrder();
return; }

19. printPreOrder Trace A C B E D
A B D C E

20. Inorder Traveral Process left subtree Process current node
Process right subtree
public void printInOrder(){
if (left != null) left.printPreOrder();
System.out.println(element.toString());
if (right != null) right.printPreOrder();
return; }

21. printInOrder Trace A C B E D
D B A C E

22. Postorder Traveral Process left subtree Process right subtree
Process current node
public void printPostOrder(){
if (left != null) left.printPreOrder();
if (right != null) right.printPreOrder();
System.out.println(element.toString());
return; }

23. printPostOrder Trace A C B E D
D B E C A

24. Methods in BinaryTree Two constructors Setter and getter for root
First takes in the element corresponding to root (left and right of root set to null)
Second does not take in any parameters (empty tree)
Setter and getter for root
size () and height () implemented as root.size() and root.height()
Traversal methods – Invoking the corresponding methods on root