Chapter 7
Non Linear Data Structure
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Linear – Data is in
sequential
Non Linear – Data
is not in sequential
Data Structure
Linear – In Sequence
• Insertion/Deletion is
possible in
linear(sequential) fashion
• Example:
–
–
–
–
Array
Stacks
Queues
Linked List
Non Linear – Not in sequence
• Insertion/Deletion is not
possible
• Example:
– Graphs
– Trees
– Hash Tree
Non-linear Data Structures
- Each node may be
connected with two
or more nodes in a
non-linear
arrangement
- Removing one of the
links could divide the
data structure into
two different data
structures
EXAMPLE OF NON-LINEAR DS
Graphs
1. It is non linear
2. Set of nodes with set
of connections
3. One-way or two-way
4. Directed (with arrow
to show direction) or
Non-Directed (two
way links without
any arrow)
Graphs Behaviour
1. Node A has three
neighbors B,C, and D
2. We will use
adjacency matrix for
graphs data
structures
Trees
• Hierarchical Data
Structures
• We have
– Nodes (root, child
nodes, leaf
nodes)
– Branches
TREES – TERMINOLOGY – TRAVERSAL – TYPE OF TREES
TREES
Definition of A Tree
• Trees are a special case of a graph data structure.
• The tree has nodes (shown with circles) that are
connected with branches. Each node will have a
parent node (except for the root) and may have
multiple child nodes.
“Trees” in real application
Trees Terminology
Branch/Edge
Child Nodes
By definition
Child of a node u
u
a
b
She has few children “a,b,c,d”
c
d
By definition
Parent node
But root node does not have parent. Poor root
u
a
b
c
d
d has a parent node “u”
By definition
subtree
“u” has a subtree of a,b,c, and d
u
a
b
c
d
By definition
Depth of a node
Level 0
u
a
e
b
f
d
c
g
h
Level 1
Level 2
By definition
Height of the tree
Level 0
u
a
e
b
f
d
c
g
Height of the tree = 2
h
Level 1
Level 2
By definition
Path Length
u
a
e
Each arrow length = 1
Thus, 4 arrows = length = 4
b
f
d
c
g
Total Length = 8 + 4 = 12
h
Each arrow length = 2
Thus, 4 arrows =
length = 8
By definition
Degree
u
“u” has a degree = 4
“a” has a degree of 0
a
e
b
f
d
c
g
h
“b” has a degree of 4
By definition
Full Tree :
u
b
e
f
c
g
This tree has a degree of 2
Height = 2
Full tree = (2^3 – 1)/(2-1) = 7
.
.
Or just calculate yourself how
much node this tree has..
h
Traversal Algorithm
We have multiple traverse patterns
Let’s go for Depth First Search
The Mr.Algo visit “u” first
Then Mr. Algo visit “b”, and
“e” and “f”.
u
b
e
f
c
g
After that, Mr. Algo back to
“c” and visit “g” and “h”
h
Traversal Algorithm
We have multiple traverse patterns
Let’s go for Breath First Search
The Mr.Algo visit “u” first
Then Mr. Algo visit “b”, and
“c”.
u
b
e
f
c
g
After that, Mr. Algo back to
“e”, “f”, “g”, and “h”
h
Rooted Tree/Ordered Tree
Rooted Tree
• Any node can become a
root for its own subnode
• General term “Free Tree”
• They have more family tree
definition such as siblings or
grandparents.
Ordered Tree
• Rooted tree where order of
each child node is specified.
[English] It is where you
determined yourself where you
put your node.
• But if we have spesific
order, then we could have
so called n-tree
(binary,quad, oct-tree)
BINARY TREE
Binary Tree
u
Left child
Right child
b
Left Subtree
e
f
c
g
Right Subtree
h
Binary Tree : Code
struct TreeNode {
int item;
//The data in this node.
TreeNode *left; //Pointer to the left subtree.
TreeNode *right;//Pointer to the right subtree.
}
Binary Tree : Code
int countNodes( TreeNode *root ) {
// Count the nodes in the binary tree to which
// root points, and return the answer.
if ( root == NULL )
return 0; // The tree is empty. It contains no nodes.
else {
int count = 1; // Start by counting the root.
count += countNodes(root->left); // Add the number of nodes
// in the left subtree.
count += countNodes(root->right); // Add the number of nodes
// in the right subtree.
return count; // Return the total.
}
} // end countNodes()
Binary Tree Traversal
Preorder traversal
sequence: F, B, A, D, C, E,
G, I, H (root, left, right)
Inorder traversal
sequence: A, B, C, D, E, F,
G, H, I (left, root, right);
note how this produces a
sorted sequence
Postorder traversal
sequence: A, C, E, D, B, H,
I, G, F (left, right, root)
DELETE A NODE
DELETE A NODE IN TREE
There are three possible cases to consider:
• Deleting a leaf (node with no children): Deleting a leaf
is easy, as we can simply remove it from the tree.
• Deleting a node with one child: Remove the node and
replace it with its child.
• Deleting a node with two children: Call the node to be
deleted N. Do not delete N. Instead, choose either its
in-order successor node or its in-order predecessor
node, R. Replace the value of N with the value of R,
then delete R.
Deleting a node with two children from a
binary search tree. The triangles represent
subtrees of arbitrary size, each with its
leftmost and rightmost child nodes at the
bottom two vertices.
Steps
• There is another simple situation: suppose the
node we're deleting has only one subtree. In
the following example, `3' has only 1 subtree.
Steps
• To delete a node with 1 subtree, we just `link
past' the node, i.e. connect the parent of the
node directly to the node's only subtree. This
always works, whether the one subtree is on
the left or on the right. Deleting `3' gives us:
Steps
• Finally, let us consider the only remaining
case: how to delete a node having two
subtrees. For example, how to delete `6'?
We'd like to do this with minimum amount of
work and disruption to the structure of the
tree.
Steps
• Erase “6”, but keep its node
Steps
• Find value that can be moved into vacated
node while preserving BST properties. Pick X
such that:
– everything in left subtree < X
– everything in right subtree > X
• If we get X from left subtree, (2) is guaranteed.
– For (1), X must be largest value in left subtree, i.e.
3.
Steps
• General algorithm: to delete N, if it has 2
subtrees, replace the value in N with the
largest value in its left subtree and then delete
the node with the largest value from its left
subtree.
• Note: the largest value has at most one
subtree.
• Why? It is the largest value: it doesn't have a
right subtree.
Let’s run sample program
• Start your Visual Studio 2010 now
• Create new C++ empty project
• Project : Binary Tree with Traversal of DFS