TORU-LOTA TREE,BST,HEAP,GRAPH

TREE

TREE IN COMPUTER SCINCE

Root

Node

Leave

Branch

Root: node without parent (A)Siblings: nodes share the same parentInternal node: node with at least one child.External node (leaf ): node without childrenAncestors of a node: parent, grandparent, grand-grandparent, etc.Descendant of a node: child, grandchild, grand-grandchild, etc.Depth of a node: number of ancestorsHeight of a tree: maximum depth of any nodeDegree of a node: the number of its childrenDegree of a tree: the maximum number of its node.

Tree Operation

BINARY SEARCH TREE

Short form of BST is Binary Search TREE

A binary search tree is a binary tree where each node has a Comparable key the key in any node is larger than the keys in all nodes in that node's left sub tree and smaller than the keys in all nodes in that node's right sub tree.

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1 5 3525

A COMPLETE BINARY TREE

What is a binary search tree?

INSERT

Insert 11

1.Start at root.

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2 10

0 5 19

4

START

INSERT

Insert 11

1.Start at root.2.Move to 10 on right, because 11>6

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2 10

0 5 19

4

INSERT

Insert 11

1.Start at root.2.Move to 10 on right, because 11>63.Move to 19 on right, because 11>10.

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2 10

0 5 19

4

INSERT

Insert 11

1.Start at root.2.Move to 10 on right, because 11>63.Move to 19 on right, because 11>10.4.Insert 11 to the left of 19, because 11<19 and right of 19 is NULL.

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2 10

0 5 19

4

INSERT

Insert 11

1.Start at root.2.Move to 10 on right, because 11>63.Move to 19 on right, because 11>10.4.Insert 11 to the left of 19, because 11<19 and right of 19 is NULL.

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2 10

0 5 19

4 11

INSERT

Insert 11

1.Start at root.2.Move to 10 on right, because 11>63.Move to 19 on right, because 11>10.4.Insert 11 to the left of 19, because 11<19 and right of 19 is NULL.

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2 10

0 5 19

4 11

11 INSERTED IN THE TREE

DELETE

Delete 5

1.Start at root and search 5.

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0 5 19

4 11

start

DELETE

Delete 5

1.Start at root and search 5.2.Move to 2 on left, because 5<6.

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2 10

0 5 19

4 11

DELETE

Delete 5

1.Start at root and search 5.2.Move to 2 on left, because 5<6.3.Move to 5 on right.

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2 10

0 5 19

4 11

DELETE

Delete 5

1.Start at root and search 5.2.Move to 2 on left, because 5<6.3.Move to 5 on right.4.Delete 5 from the right of 2.

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2 10

0 19

4 11

DELETE

Delete 5

1.Start at root and search 5.2.Move to 2 on left, because 5<6.3.Move to 5 on right.4.Delete 5 from the right of 2.5.Now take 4 and set at the right of 2.

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2 10

0 4 19

11

DELETE

Delete 5

1.Start at root and search 5.2.Move to 2 on left, because 5<6.3.Move to 5 on right.4.Delete 5 from the right of 2.5.Now take 4 and set at the right of 2.

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2 10

0 4 19

11

5 DELETED FROM THE TREE

TRAVERSINGPreorder:

1.Visit the Root2.Traverse the Left subtree

3.Traverse The Right subtree

7>1>0>3>2>5>4>6>9>8>10

In order:1.Traverse the Left subtree.

2.Visit the Root.3.Traverse the Right subtree.

Post Order :1.Traverse the Left subtree

2. Traverse the Right subtree.3.Visit the Root.

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0 38 10

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7,1,0,3,2,5,4,6,9,8,10

HeapsA heap is a certain kind of complete binary tree.

Each node in a heapcontains a key thatcan be compared toother nodes' keys.

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Types of Heaps:• There are two types of heaps

1. Max Heap2. Min Heap

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Max Heap

The “MAX heap property"

requires that eachnode's key is >= thekeys of its children

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Min Heap

The “Min heap property"

requires that eachnode's key is <= thekeys of its children

Adding a Node to a Heap

Put the new node in the next available spot.

Push the new node upward, swapping with its parent until the new node reaches an acceptable location.

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Adding a Node to a Heap

Put the new node in the next available spot.

Push the new node upward, swapping with its parent until the new node reaches an acceptable location.

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Adding a Node to a Heap

Put the new node in the next available spot.

Push the new node upward, swapping with its parent until the new node reaches an acceptable location.

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Adding a Node to a Heap

The parent has a key that is >= new node, or

The node reaches the root.

The process of pushing the new node upward is called reheapification upward. 19

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Removing the Top of a Heap

Move the last node onto the root.

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Removing the Top of a Heap

Move the last node onto the root.

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Removing the Top of a Heap

Move the last node onto the root.

Push the out-of-place node downward, swapping with its larger child until the new node reaches an acceptable location.

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Removing the Top of a Heap

Move the last node onto the root.

Push the out-of-place node downward, swapping with its larger child until the new node reaches an acceptable location.

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4222135

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42

27

Removing the Top of a Heap

Move the last node onto the root.

Push the out-of-place node downward, swapping with its larger child until the new node reaches an acceptable location.

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42

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Removing the Top of a Heap

The children all have keys <= the out-of-place node, or

The node reaches the leaf.

The process of pushing the new node downward is called reheapification downward.

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Implementing a Heap

We will store the data from the nodes in a partially-filled array.

An array of data

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Implementing a Heap

• Data from the root goes in the first location of the array.

An array of data

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Implementing a Heap

• Data from the next row goes in the next two array locations.

An array of data

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Implementing a Heap

• Data from the next row goes in the next two array locations.

An array of data

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Implementing a Heap

• Data from the next row goes in the next two array locations.

An array of data

2127

23

42

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42 35 23 27 21

We don't care what's inthis part of the array.

GraphA graph that consists of a set of nodes (vertices) and a set of edges that relate the nodes to each other.

Degree: The number of edge of a node is called degree of that node.

Connected Graph: A graph said to be connected if and only if there is a simple path between two nodes.

Complete Graph: A graph is said to be complete if every node is adjacent to every other node.

Tree Graph: A connected graph without any cycle is called a tree graph or free tree or only tree.

Cycle: Cycle is a closed simple path with length three or more.

Here,ACDA is a cycle in this Graph.

Labeled Graph: A graph is said to be labeled if its edegs are assigned data.

Weighted Graph: A graph is said to be weighted if each edge is assigned a non-negative value called the weight of length.

GRAPHMULTIPLE EDGE / DISTINCT EDGE: If two edge connected the same end points.

LOOP : An edge is called a loop if it has same identical end points.

DIRECTED GRAPG

DIRECTED GRAPH : In a graph the vertices that are connected together , where all the edges are directed from one two another vertices .From this graph v1 is initial points and v2 is terminal points.v1 is a predecessor of v2 and v2 is successer of v1 .

DEGREE : Directed graph has two types degree .• Indegree• Outdegree

DIRECTED GRAPH

SOURCE : A node is called as a source if it has a positive out degree and 0 indegree .

SINK : A node is called as a sink if it has a positive indegree and 0 out degree .

ADJACENT NODE : Two nodes are adjacent if they connected via an edge .

ADJANCENT MATRIX : Connectivity between two nodes can be tested quickly.

ADJACENT

ADJACENTADJACENT LIST : Adjacent node can be represented also adjacent list.

• 0 -> 1• 1 -> 2 , 3• 2 -> • 3 -> 0 , 2

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