Data Structures and Algorithms
Lecture 15 and 16 (BinaryTrees)
Instructor: Quratulain
Date: 3 and 6 October, 2009
Faculty of Computer Science, IBA
Introduction
Tree is a non-linear structure.
 In computer science Tree is a ADT of
hierarchical structure.
 Tree is divided into levels from root to
leaf nodes.

Tree Terminology
root
A
child
sibling
parent
B
C
D
interior (internal) node
E
F
G
H
leaf node
I
J
subtree
The height of a tree is the maximum level in the tree.
Terminology

Node, branch, root, indegree, outdegree, leaf, parent,
child, siblings, ancestor, descendent, path, level, height,
depth, subtree
Binary Tree

Each node has 0, 1 and 2 childs

Each node of a binary tree defines a left
and a right subtree. Each subtree is itself a
tree.
Height of a Binary Tree

The height of a binary tree is the length of the
longest path from the root to a leaf node. Let
TN be the subtree with root N and TL and TR be
the roots of the left and right subtrees of N.
Then
height(N) = height(TN) =
{
-1
1+max( height(TL), height(TR))
if TN is empty
if TN not empty
© 2005 Pearson Education, Inc., Upper
Saddle River, NJ. All rights reserved.
Height of a Binary Tree (concluded)
Degenerate binary tree
© 2005 Pearson Education,
Inc., Upper Saddle River,
NJ. All rights reserved.
Types Binary Trees
Complete binary tree: A complete binary
tree of height h has all possible nodes
through level h-1, and the nodes on depth
h exist left to right with no gaps.
 Full binary tree: A tree in which every node
other than the leaves has two children
 Perfect binary tree: A full binary tree in
which all leaves are at the same same level

Binary tree
Total number of nodes in complete binary
tree is from 2h – 1 to 2h+1
 The height of complete binary tre
h=(log2n)

Some valid Binary Search
Trees
Some invalid Binary Search
Trees
Application of Binary Trees
For two-way decisions at each point in a
process. Then the number of comparison
could be reduced.
 to path finding, connected components.
 Application of Sorting
 Application of searching
 Expression tree

Linklist and Arrays for binary tree
Store in array using formula:
2n+1 for left child
2n+2 for right child
Binary Trees: Traversals
• There are three classic ways to traverse a
tree: NLR, LNR and LRN
Preorder (NLR) Traversal
preorder(node)
{
if (node is not null)
process(node)
preorder(node->left)
preorder(node->right)
}
Inorder (LNR) Traversal
inorder(node)
{
if (node is not null)
inorder(node->left)
process(node)
inorder(node->right)
}
Postorder (LRN) Traversal
postorder(node)
{
if (node is not null)
inorder(node->left)
inorder(node->right)
process(node)
}
Traversing Binary Search
Trees
• Preorder: 23 18 12 20 44 35 52
• Postorder: 12 20 18 35 52 44 23
• Inorder: 12 18 20 23 44 35 52
Delete to Binary Search Trees

The deletion algorithm is more complicated, because
after a non-leaf node is deleted, the "hole" in the
structure needs to be filled by a leaf node. There are
three possibilities:
◦ To delete a leaf node (no children): disconnect it.
◦ To delete a node with one child: bypass the node and directly
connect to the child.
◦ To delete a node with two children: find the smallest node in its
right subtree (or the largest node in its left subtree), copy the
minimum value into the info field of the "node to be deleted"
and then delete the minimum node instead, which can only
have a right child, so the situation becomes one of the above
two.
Binary Search Trees vs. Arrays
Same O(log2N) search
 Better insertion time: O(log2N) vs. O(N)
 Better deletion
 What is worse? O(N)
 BST requires more space - 2 references
for each data element

Huffman Codes
Binary trees can be used in an interesting
way to construct minimal length
encodings for messages when the
frequency of letters used in the messages
is known.
 A special kind of binary tree, called a
Huffman coding tree is used to accomplish
this.

Huffman Codes
Huffman is a coding algorithm presented by
David Huffman in 1952. It's an algorithm which
works with integer length codes.
 Huffman is the best option because it's
optimal.
 The position of the symbol depends on its
probability. Then it assigns a code based on its
position in the tree. The codes have the prefix
property and are instantaneously decodable
thus they are well suited for compression and
decompression.

Example
suppose we know that the frequency
of occurrences for six letters in a
message are as given below:
 To build the Huffman tree, we sort
the frequencies into increasing order
(4, 5, 7, 8, 12, 29). Then we choose the
two smallest values, 4 and 5, and
construct a binary tree with labeled
edges:

E
29
I
5
N
7
P
12
S
4
T
8
Next, we replace the two
smallest values 4 and 5 with
their sum, getting a new
sequence, (7, 8, 9, 12, 29). We
again take the two smallest
values and construct a labeled
binary tree.
 We now have the frequencies
(15, 9, 12, 29) which must be
sorted into (9, 12, 15, 29) and
the two lowest are selected
once again.
 Now, we combine the two
lowest which are 15 and 21 to
give the tree.


The two remaining frequencies, 36 and 29, are
now combined into the final tree. Notice that
it does not make any difference which one is
placed as the left subtree and which in the
right subtree.




From this final tree, we find the encoding
for this alphabet
Using this code, a message like SENT
would be coded as 01111000001
If the message had been coded in the
"normal" way, each letter would have
required 3 bits. The entire message is 65
characters long so 195 bits would be
needed to code the message (3*65).
Using the Huffman code, the message
requires
1*29+4*5+3*12+3*7+4*4+3*8=146 bits.
This code can be applied to the English
Language by using average frequency
counts for the letters.
E
1
I
0110
P
010
N
000
S
0111
T
001