1. Tree Traversal

2. Pre Order Tree Traversal
Visit the root
Traverse the left sub-tree,
Traverse the right sub-tree

3. In Order Tree Traversal
Traverse the left sub-tree,
Visit the root
Traverse the right sub-tree

4. Post Order Tree Traversal
Traverse the left sub-tree,
Traverse the right sub-tree
Visit the root

5. Post Order Tree Traversal
A*(((B+C)*(D*E))+F)
A B C + D E * * F + *

6. Tree Traversal

7. Example Print the pre, in and post order form of the following tree
DAICBHEGF – in
HBDIACEFG – pre
ACIDBGFEH – post H E B F D I G A C

8. Example class Node { private Object value; private Node left, right;
public Node (Object value) {
this.value = value; }
...

9. Example class Tree { private Node root; public Tree () { root = null;}
...

10. In Order traversal public void inOrder () { inOrder (root); }
private void inOrder (Node current) {
if (current != null) {
inOrder(current.getLeft());
visit(current);
inOrder(current.getRight());

11. Pre Order traversal public void preOrder () { preOrder (root); }
private void preOrder (Node current) {
if (current != null) {
visit(current);
preOrder(current.getLeft());
preOrder(current.getRight());

12. Post Order traversal public void postOrder () { postOrder (root); }
private void postOrder (Node current) {
if (current != null) {
postOrder(current.getLeft()); postOrder(current.getRight());
visit(current);

13. Example Can we reconstruct a tree from it’s InOrder traversal ?
Are there two different trees with the same InOrder traversal
What about pre and post order ? D A

14. Example
Can we reconstruct a tree given both it’s in order and pre order traversals ?
in order - DAICBHEGF
pre order - HBDIACEFG (find the root)
Left sub tree DAICB
BDIAC
Right sub tree EGF
EFG

15. Reconstructing a binary tree
ReconstructTree (inOrder in[] preOrder pre[])
if (n==0)
return null
root (T)  pre[1]
find index i s.t in[i] = pre[1]
left(T)  Reconstruct (in[1…i-1], pre [2…i])
right(T)  Reconstruct (in[i+1..n], pre [i+1…n])
return T

16. Reconstructing a binary tree
Time complexity
This is the same recurrence relation of quick sort

17. Binary Search Trees
In a BST the in order output is a sorted list of the tree nodes.
If we obtain a pre order path of a BST, we can sort the nodes and use the output to reconstruct the tree using the previous algorithm

18. Reconstructing a binary tree
ReconstructTree (inOrder in[] postOrder post[])
if (n==0)
return null
root (T)  post[n]
find index i s.t in[i] = post[n]
left(T)  Reconstruct (in[1…i-1], post [1…i-1])
right(T)  Reconstruct (in[i+1..n], post[i…n-1])
return T

19. Level traversal (BFS) public void BFS () { Queue q = new Queue ();
if {root != null)
q.enqueue (root);
while (! q.isEmpty()) {
Node current = (Node)queue.dequeue();
visit (current);
if (current.getLeft() != null) q.enqueue(current.getLeft());
if (current.getRight() != null) q.enqueue(current.getRight()); }

20. Depth private int depth (Node node) { if (current == null) { return 0;}
else {
return Math.max(depth(node.getLeft()), depth(node.getRight())) + 1;

21. private void printLevel (Node current, int level) {
if (current != null) {
if (level == 0) {
visit (current); }
else {
printLevel (current.getLeft(), level - 1);
printLevel (current.getRight(), level - 1);
public void printAllLevels () {
int depth = depth (root);
for (int i = 0; i < depth; i++);
printLevel(root, i);

22. Questions
1. Draw a single Binary Tree such that each node contains a single character and:
a.  The pre-order traversal results in EXAMFUN
b.  The in-order traversal results in MAFXUEN

23. Questions
2. An in-order traversal of a Binary search tree yields a sorted list of elements in O(n) time. Does this fact contradict our comparison-based lower bound of nlogn time? Can something similar be done using a binary heap?

24. Questions
3. Is there a heap T storing seven distinct elements such that a preorder traversal of T yields the elements of T in sorted order?
How about an in-order traversal?
How about a post-order traversal?
How about a post-order traversal to yield the elements in reverse sorted order?

25. Questions
4. Let T be a binary search tree with more than a single node. Is it possible that the preorder traversal of T visits the nodes in the same order as the post-order traversal of T?
If so, give an example; otherwise, argue why this cannot occur. Likewise, is it possible that the preorder traversal of T visits the nodes in the reverse order of the post-order traversal of T?
If so, give an example; otherwise, argue why this cannot occur.

26. Questions
5. Given the following binary tree, print out the order of the nodes for an in-order traversal, preorder traversal, and post-order traversal.
/ (division operator)
/ \
/ \ *
/ \ / \
/ \ a
^ *
/ \ / \
b 2 * c
/ \
4 a

27. Questions
6. Write a method that given a TreeNode N , returned the number of elements contained at the tree rooted by N.