Data Structure and Algorithms Binary Tree

Binary trees: introduction (complete and
extended binary trees),
Memory representation (sequential, linked)
Binary tree traversal: pre-order, in-order and
post-order (traversal algorithms using stacks)

A tree is a finite set of zero or more nodes
such that:
There is a specially designated node called
the root.
The remaining nodes are partitioned into
n>=0 disjoint sets T1, ..., Tn, where each of
these sets is a tree.
We call T1, ..., Tn the subtrees of the root.

Level and Depth
K L
E F
B
G
C
M
H I J
D
A
Level
0
1
2
3
node (13)
degree of a node(shown by green
number)
leaf (terminal)
nonterminal
parent
children
sibling
ancestor
descendant
level of a node
height(depth) of a tree (4)
3
2 1 3
2 0 0 1 0 0
0 0 0

Terminology
The degree of a node is the number of subtrees
of the node
The degree of A is 3; the degree of C is 1.
The node with degree 0 is a leaf or terminal
node.
A node that has subtrees is the parent of the
roots of the subtrees.
The roots of these subtrees are the children of
the node.
Children of the same parent are siblings.
The ancestors of a node are all the nodes
along the path from the root to the node.

Tree Properties
A
B C
D
G
E F
IH
Property Value
Number of nodes
Height
Root Node
Leaves
Interior nodes
Number of levels
Ancestors of H
Descendants of B
Siblings of E
degree of node A

Binary Trees
A special class of trees: max degree for each
node is 2
Recursive definition: A binary tree is a finite
set of nodes that is either empty or consists of
a root and two disjoint binary trees called the
left subtree and the right subtree.

Samples of Trees
A
B
A
B
A
B C
GE
I
D
H
F
Complete Binary Tree
Skewed Binary Tree
E
C
D

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

Full BT vs. Complete BT
A full binary tree of depth k is a binary tree of depth k
having 2k
-1 nodes, k>=1.
A binary tree with n nodes and depth k is
complete iff its nodes correspond to the nodes numbered
from 1 to n in the full binary tree of depth k.
A
B C
GE
I
D
H
F
A
B C
GE
K
D
J
F
IH ONML
Full binary tree of depth 4Complete 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 2i if 2i<=n. If 2i>n, then i has no
left child.
rightChild(i) is at 2i+1 if 2i+1<=n. If 2i+1>n,
then i has no right child.

Extended Binary TreeA binary tree T is said to be a 2-tree or an extended binary tree if each
node N has either 0 or 2 children.
The nodes with 2 children are called internal nodes.
The nodes with o children are called external nodes. A
B C
G
D F
A
B C
G
D F
Fig: Binary Tree T Fig: Extended 2-tree

Sequential
RepresentationA
B
--
C
--
--
--
D
--
.
E
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
.
[16]
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
A
B
C
D
E
F
G
H
I
A
B
E
C
D
A
B C
GE
I
D
H
F
(1) waste space
(2) insertion/deletion
problem

Linked Representation
struct btnode {
int data;
btnode *left, *right;
};
dataleft right
data
left right

Binary Tree Traversals
There are three standard ways of traversing a
binary tree T with root R.
These three algorithms, called:
preorder
inorder
postorder

Arithmetic Expression Using BT
+
*
A
*
/
E
D
C
B
preorder traversal
+ * * / A B C D E
prefix expression
inorder traversal
A / B * C * D + E
infix expression
postorder traversal
A B / C * D * E +
postfix expression

Preorder Traversal (recursive
version)
Algorithm:
1. Process the root R.
2. Traverse the left subtree of R in preorder.
3. Traverse the right subtree of R in preorder.
Pseudo-Code:
void preorder(btnode ptr)
/* preorder tree traversal */
{
if (ptr!=NULL) {
cout<<ptr->data;
preorder(ptr->left);
predorder(ptr->right);
}
}
+ * * / A B C D E

Inorder Traversal (recursive version)
Algorithm:
1. Traverse the left subtree of R in inorder.
3. Traverse the right subtree of R in inorder.
Pseudo-Code:
void inorder(btnode ptr)
/* inorder tree traversal */
{
if (ptr!=NULL) {
inorder(ptr->left);
cout<<ptr->data;
indorder(ptr->right);
}
}
A / B * C * D + E

Postorder Traversal (recursive version)
Algorithm:
1. Traverse the left subtree of R in postorder.
2. Traverse the right subtree of R in postorder.
Pseudo-Code:
void postorder(btnode ptr)
/* postorder tree traversal */
{
if (ptr!=NULL) {
postorder(ptr->left);
postorder(ptr->right);
cout<<ptr->data;
}
}
A B / C * D * E +

Preorder Traversal (using stack)
PREORD(INFO, LEFT, RIGHT, ROOT)
1. Set TOP:= 1, STACK[1]:= NULL and PTR := ROOT.
2. Repeat Steps 3 to 5 while PTR != NULL:
3. Apply PROCESS to INFO[PTR].
4. If RIGHT[PTR]!= NULL then
Set TOP := TOP+1, and STACK[TOP]:= RIGHT[PTR].
5. If LEFT[PTR]!= NULL then
Set PTR:= LEFT[PTR].
Else
Set PTR:= STACK[TOP] TOP:= TOP-1.
[End of If structure]
[End of step 2 Loop]
6. Exit.
+ * * / A B C D E

Inorder Traversal (using stack)
INORD(INFO, LEFT, RIGHT, ROOT)
1.Set TOP:= 1, STACK[1]:= NULL and PTR := ROOT.
2.Repeat while PTR != NULL:
a) Set TOP:=TOP+1, STACK[TOP]:=PTR.
b) Set PTR:= LEFT[PTR].
[End of loop]
3.Set PTR:=STACK[TOP], TOP:=TOP-1.
4.Repeat Steps 5 to 7 while PTR != NULL:
5.Apply PROCESS to INFO[PTR].
6.If RIGHT[PTR]!= NULL, then:
a) Set PTR:=RIGHT[PTR]
b) Go to Step 2.
7.Set PTR:=STACK[TOP] and TOP:=TOP-1.
[End of step 4 Loop.]
8.Exit
A / B * C * D + E

Postorder Traversal (using stack)
POSTORD(INFO, LEFT, RIGHT, ROOT)
1.Set TOP:= 1, STACK[1]:= NULL and PTR := ROOT.
2.Repeat Step 3 to 5 while PTR!=NULL:
3.Set TOP := TOP+1, STACK[Top]:=PTR.
4.If RIGHT[PTR]!=NULL, then:
Set TOP := TOP+1, Stack[Top]:=-
RIGHT[PTR].
5.Set PTR:=LEFT[PTR]
[End of step 3 Loop]
6.Set PTR:=STACK[Top], TOP := TOP-1.
7.Repeat while PTR>0:
a)Apply PROCESS to INFO[PTR].
b)PTR:=STACK[Top], TOP := TOP-1.
[End of Loop]
8.Repeat while PTR<0:
a) Set PTR:= -PTR.
b) Go to Step 2.
A B / C * D * E +

Data Structure and Algorithms Binary Tree

Data Structure and Algorithms Binary Tree

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