Complete binary tree the tree is said to be complete

COMPLETE BINARY TREE
The tree is said to be complete binary tree if all its levels, except
possibly the last, have the maximum number of possible nodes
and if all the nodes at last level appear as far left as possible.
Maximum number of nodes at any level r =2
r
Eq:- at level 0 maximum number of nodes=2º=1
At level 2 maximum number of nodes=2²=4 and so on.
.
12

COMPLETE BINARY TREE
Suppose given a complete binary tree having 26 nodes. We are labeling the
nodes from 1 to 26.
Left child of node k=2*K
Right child of node k=2*K+1
Parent of node K=[K/2]
Depth of complete binary tree T
n
with n node
D
n
= [log
2
n +1]
13

FULL BINARY TREE VS COMPLETE
BINARY TREE
14

FULL VS. COMPLETE BINARY TREE
15

Full BT vs. Complete BT
A full binary tree of depth k is a binary tree of
depth k having 2
k
-1 nodes, k >=1.
A binary tree with n nodes and depth k is
complete if its nodes correspond to the nodes
numbered from 1 to n in the full binary tree of
depth k .
A
B
C
G
E
I
D
H
F
A
B
C
G E
K
D
J
F
I H O N M
L
Full binary tree of depth 4
Complete binary tree

Complete Binary Tree
If a complete binary tree with n nodes
(depth=└log n + 1┘)
is represented sequentially,
then for any node with index i , 1<= i <= n , we
have:
parent ( i ) is at └i /2┘ if i !=1. If i =1, i is at the
root and
has no parent.
leftChild ( i ) is at 2 i if 2 i <= n . If 2 i >n, then i has
no
left child.
rightChild ( i ) is at 2 i +1 if 2 i +1<= n . If 2 i +1>n,

EXTENDED BINARY TREE
•
An extended binary tree is a transformation of any binary tree
into a complete binary tree. This transformation consists of
replacing every null subtree of the original tree with ‘special
nodes.'
•
The nodes from the original tree are then internal nodes, while
the ``special nodes'' are external nodes.
•
For instance, consider the following binary tree in fig(a). Fig
(b) is the extended binary tree.
•
Empty circles represent internal nodes, and filled circles
represent external nodes. Every internal node in the extended
tree has exactly two children, and every external node is a leaf.
The result is a complete binary tree.
18

Maximum Number of Nodes
in BT
The maximum number of nodes on level
i of a binary tree is 2
i
, i>=0.
The maximum number of nodes in a
binary tree
of depth k is 2
k
-1 , k>=1.

EXAMPLE
a
b
20

REPRESENTATION OF BINARY TREES IN
MEMORY
.

Array Representation

Linked Representation
21

ARRAY REPRESENTATION
.
•
Tree Nodes are stored in the array.
•
Root node is stored at index ‘0’
•
If a node is at a location ‘i’, then its

left child is located at 2 * i + 1

right child is located at 2 * i + 2
•
Root node is stored at index ‘1’
•
If a node is at a location ‘i’, then its

left child is located at 2 * i

right child is located at 2 * i + 1
•
This storage is efficient when it is a complete binary tree,
because a lot of memory is wasted.
22

EXAMPLE 1
23

EXAMPLE 2
Disadvantages
(1) waste space
(2) insertion/deletion
problem
24

LINKED REPRESENTATION
•
If a binary tree is not complete , a better choice for storing
it is to use a linked representation.
•
It uses three array INFO, RIGHT, LEFT, and a pointer
variable ROOT as follows.