Tree Traversal

Pre Order Tree Traversal

1.Visit the root

2.Traverse the left sub-tree,

3.Traverse the right sub-tree

In Order Tree Traversal

1. Traverse the left sub-tree,

2. Visit the root

3. Traverse the right sub-tree

Post Order Tree Traversal

1. Traverse the left sub-tree,

2. Traverse the right sub-tree

3. Visit the root

Post Order Tree Traversal

• A*(((B+C)*(D*E))+F)

• A B C + D E * * F + *

Tree Traversal

Example

• Print the pre, in and post order form of the following tree

DAICBHEGF – in

HBDIACEFG – pre

ACIDBGFEH – post

H

EB

D F

I G

CA

Example

class Node { private Object value; private Node left, right;

public Node (Object value) { this.value = value;

} ...

}

Example

class Tree {

private Node root;

public Tree () {

root = null;

}

...

}

In Order traversal

public void inOrder () { inOrder (root);

}private void inOrder (Node current) {if (current != null) {

inOrder(current.getLeft());visit(current);inOrder(current.getRight());

} }

Pre Order traversal

public void preOrder () { preOrder (root);

}private void preOrder (Node current) { if (current != null) {

visit(current);preOrder(current.getLeft());preOrder(current.getRight());

}}

Post Order traversal

public void postOrder () { postOrder (root);

}private void postOrder (Node current) {if (current != null) {

postOrder(current.getLeft()); postOrder(current.getRight());

visit(current); }

}

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

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

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

Reconstructing a binary tree

• Time complexity

• This is the same recurrence relation of quick sort

( ) ( 1) ( ) ( )T n T i T n i O n

2( ) ( )

( ) ( log )

T n n worst case

T n n n averagecase

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

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

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());}

}

Depth

private int depth (Node node) { if (current == null) {

return 0; }else {

return Math.max(depth(node.getLeft()), depth(node.getRight())) + 1;

} }

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); }

}

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

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?

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?

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.

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) / \

/ \ - * / \ / \ / \ 2 a ^ * / \ / \ b 2 * c / \ 4 a

Questions

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