# Why binary Trees?

```Data Structures and
Algorithms
Lec(5) Binary Tree
What is a Tree?
• A tree, is a finite set of nodes together with a finite set of directed
edges that define parent-child relationships. Each directed edge
connects a parent to its child. Example:
A
Nodes={A,B,C,D,E,f,G,H}
Edges={(A,B),(A,E),(B,F),(B,G),(B,H),
(E,C),(E,D)}
E
D
B
C
F
H
G
2
What is a Tree? (contd.)
•
A tree satisfies the following properties:
1.
2.
3.
4.
It has one designated node, called the root, that has no parent.
Every node, except the root, has exactly one parent.
A node may have zero or more children.
There is a unique directed path from the root to each node.
5
5
3
3
2
4
tree
1
6
5
2
4
1
Not a tree
3
6
2
4
Not a tree
1
6
3
Tree Terminology (Contd.)
• Degree: The number of subtrees of a node
– Each of node D and B
has degree 1.
– Each of node A and E
has degree 2.
– Node C has degree 3.
– Each of node F,G,H,I,J has degree 0.
An Ordered Tree
with size of 10
Siblings of A
A
B
D
H
C
E
I
F
G
J
Siblings
of E
•
•
•
•
Leaf: A node with degree 0.
Internal or interior node: a node with degree greater than 0.
Siblings: Nodes that have the same parent.
Size: The number of nodes in a tree.
4
Tree Terminology (Contd.)
• Level (or depth) of a node v: The length of the path from the root to v.
• Height of a node v: The length of the longest path from v to a leaf
node.
– The height of a tree is the height of its root mode.
– By definition the height of an empty tree is -1.
• The height of the tree is 4.
Level 0
A
• The height of node C is 3.
B
D
H
E F
I
Level 1
C
G
Level 2
Level 3
J
k
Level 4
5
Why Trees?
• Trees are very important data structures in computing.
• They are suitable for:
– Hierarchical structure representation, e.g.,
• File directory.
• Organizational structure of an institution.
• Class inheritance tree.
– Problem representation, e.g.,
• Expression tree.
• Decision tree.
– Efficient algorithmic solutions, e.g.,
• Search trees.
6
N-ary Trees
•
An N-ary tree is an ordered tree that is either:
1. Empty, or
2. It consists of a root node and at most N non-empty N-ary
subtrees.
It follows that the degree of each node in an N-ary tree is at most N.
Example of N-ary trees:
•
•
B
5
2
9
7
5
2-ary (binary) tree
D
C
G
D
B
J
E
F
3-ary (tertiary)tree
A
7
Binary Trees
•
•
A binary tree is an N-ary tree for which N = 2.
Thus, a binary tree is either:
1. An empty tree, or
2. A tree consisting of a root node and at most two non-empty
binary subtrees.
Example:
5
2
9
7
5
8
Binary Trees (Contd.)
• Example showing the growth of a complete binary tree:
9
Binary Trees Implementation
left
key
right
Example: A binary tree representing a + (b - c) * d
+
a
*
-
b
d
c
10
Application of Binary Trees
•
Binary trees have many important uses. Two examples are:
1. Binary decision trees.
• Internal nodes are conditions. Leaf nodes denote decisions.
false
Condition1
True
decision1
Condition2
false
True
decision2
Condition3
false
decision3
•
True
decision4
+
Expression Trees
a
*
-
b
d
c
11
It is an ordered tree with the following properties
1. Each node can have at most two subtrees
2. Each subtree is identified as being either left
subtree or the right subtree of its parent
3. It may be empty
Note:
 Property 1 says that each node can have maximum two
children
 The order between the children of a node is specified by
labeling its children as left child and right child
Why binary Trees?
Binary trees naturally arise in many different applications
1. Expression Tree
- A central data structure in compiler design
- Interior nodes contain operators and the leaf nodes have operands
- An expression is evaluated by applying the operator at root to the
values obtained by recursively evaluating the left and right subtrees
- The following tree corresponds the expression: (a+((b-c)*d)
+
*
a
b
d
c
Why binary Trees?
2. Huffman Coding Tree
- Its is used in a simple but effective data compression algorithm
- In this tree, each symbol is stored at a leaf node
- To generate the code of a symbol traverse the tree from the root to
the leaf node that contains the symbol such that each left link
corresponds to 0 and each right link corresponds to 1
- The following tree encodes the symbols a, b, c, d
0
1
a
1
0
0
b
d
1
c
The code of a is 0, and that of b is 100
Recursive definition of Binary Tree
A binary tree is
- empty OR
- a node, called the root node, together with two binary
trees, which are disjoint from each other and the root
node. These are called left and right subtrees of the root
Types of Binary Tree
Full - Every node has exactly two children in all levels,
except the last level. Nodes in last level have 0
children
Complete - Full up to second last level and last level is
filled from left to right
Other - not full or complete
Full,
Complete
or Other?
B
I
J
K
not binary
A
C
H
L
M
T
S
D
G
F
E
O
N
R
Q
P
Full,
Complete
or other?
A
B
I
K
D
H
L
M
T
S
G
F
O
N
R
P
Full,
Complete
or other?
A
B
G
H
I
K
D
L
T
F
O
N
M
S
R
P
Full,
Complete
or other?
A
B
T
L
S
G
H
I
K
D
M
F
O
N
R
P
Full,
Complete
or other?
A
B
T
L
S
G
H
I
K
D
M
N
O
F
R
P
Full,
Complete
or Other?
A
T
L
S
G
H
I
K
D
B
M
N
O
P
F
Q
R
Full,
Complete
or other?
A
G
H
I
K
D
B
L
M
O
N
T
P
S
F
Q
R
Full,
Complete
or Other?
A
G
H
I
K
D
B
L
M
N
O
P
F
Q
R
Binary Tree Traversal
Process
To process a node means to perform some simple operation like printing the
contents of the node or updating the contents of the node
Traversal
To traverse a finite collection of objects means to process each object in the
collection exactly once
Examples
1. List traversal – to process each element of list exactly once
2. Tree traversal – to process each node of tree exactly once
Traversal Of Binary Tree
There are three methods for the traversal of a binary tree
1. Preorder Traversal
Each node is processed before any node in either of its subtrees
2. Inorder Traversal
Each node is processed after all nodes in its left subtree and before any node in
its right subtree
3. Postorder Traversal
Each node is processed after all nodes in both of its subtrees
Preorder Traversals
Visit the root
Visit Left subtree
Visit Right subtree
.1
.2
.3
1
A
B
2
G
12
H
8
3I
K
4
11
D
5
L
T
6
F
13
O
14
N
10
M
9
S7
R
15
P
16
Algorithm TraversePreorder(n)
Process node n
if n is an internal node then
TraversePreorder( n -> leftChild)
TraversePreorder( n -> rightChild)
Inorder Traversals
Visit Left subtree
Visit the root
Visit Right subtree
.1
.2
.3
10
1A
6B
1
G
11
8
H
2I
1
K
1
12
D
4
L
T
3
16
F
14
O
N9
M
7
5
S
R
13
P
15
Algorithm TraverseInorder(n)
if n is an internal node then
TraverseInorder( n -> leftChild)
Process node n
if n is an internal node then
TraverseInorder( n -> rightChild)
Postorder Traversals
Visit Left subtree
Visit Right subtree
Visit the root
.1
.2
.3
A
16
B
9
G
10
H
8
2
5I
K
1
D
15
L
2
34
T
2
14
F
O
13
N
7
M
6
S
3
R
11
P
12
Algorithm TraversePostorder(n)
if n is an internal node then
TraversePostorder( n -> leftChild)
TraversePostorder( n -> rightChild)
Process node n
Elements: Any data type
Structure: A binary tree either is empty OR a node, called the root node,
together with two binary trees, which are disjoint from each
other and the root node. These are called left and right
subtrees of the root
Operations:
1/Creating a Node:
struct TreeNode
{
int value;
TreeNode *left;
TreeNode *right;
};
2/insert(int num);delete(int num);
3/Node *search(int num);
4/preorderWalk( Node *root );
5/inorderWalk( Node *root );
6/postorderWalk
Height and the number of nodes in a Binary Tree
 If the height of a binary tree is h then the maximum number of nodes in the tree
is 2h -1
 If the number of nodes in a complete binary tree is n, then
2h - 1 = n
2h = n + 1
h = log(n+1)  O(logn)
Binary Search Trees(BST)
6
<
2
1
9
>
4 =
Binary Search Trees
8
36
Binary Search Tree
• A binary search tree is a
binary tree storing keys
(or key-value entries) at
its internal nodes and
satisfying the following
property:
– Let u, v, and w be three
nodes such that u is in the
left subtree of v and w is in
the right subtree of v. We
have
key(u)  key(v)  key(w)
6
2
1
• External nodes do not
store items
37
Binary Search Trees
9
4
8
Search
• To search for a key k, we
trace a downward path
starting at the root
• The next node visited
depends on the outcome
of the comparison of k
with the key of the
current node
• If we reach a leaf, the key
return nukk
• Example: find(4):
Algorithm TreeSearch(k, v)
if T.isExternal (v)
return v
if k < key(v)
return TreeSearch(k, T.left(v))
else if k = key(v)
return v
else { k > key(v) }
return TreeSearch(k, T.right(v))
<
2
– Call TreeSearch(4,root)
1
O(Log2 N)
38
Binary Search Trees
6
9
>
4 =
8
Insertion
6
<
• To perform operation inser(k,
o), we search for key k (using
TreeSearch)
• Assume k is not already in the
tree, and let let w be the leaf
reached by the search
• We insert k at node w and
expand w into an internal
node
• Example: insert 5
2
9
>
1
4
8
>
w
6
2
1
9
4
8
w
O(Log2 N)
39
5
Binary Search Trees
Deletion
To perform operation
remove(k), we search for key k
Assume key k is in the tree,
and let let v be the node
storing k
If node v has a leaf child w, we
remove v and w from the tree
with operation
removeExternal(w), which
removes w and its parent
Example: remove 4
2
•
9
>
4 v
1
8
w
5
•
6
2
•
1
40
6
<
•
Binary Search Trees
9
5
8
Deletion (cont.)
• We consider the case where the
key k to be removed is stored at
a node v whose children are
both internal
– we find the internal node w
that follows v in an inorder
traversal
– we copy key(w) into node v
– we remove node w and its left
child z (which must be a leaf)
by means of operation
removeExternal(z)
• Example: remove 3
v
3
2
8
6
w
Binary Search Trees
9
5
z
1
v
5
2
8
6
O(Log2 N)
41
1
9
Summary: Trees Data Structures

Tree




Nodes
Each node can have 0 or more children
A node can have at most one parent
Binary tree

Tree with 0–2 children per node
Tree
Binary Tree
Trees

Terminology




Root  no parent
Leaf  no child
Interior  non-leaf
Height  distance from root to leaf
Root node
Interior nodes
Leaf nodes
Height


A binary tree is a non-linear linked list where
each node may point to two other nodes.
Binary trees are excellent data structures for
searching large amounts of information. They
are commonly used in database applications
to organize key values that index database
records.
```