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What are the properties of a tree in a network graph?
1. It consists of all the nodes of the graph.
2. If the graph has N number of nodes, the tree will have (N – 1) branches.
3. There will be only one closed path in the tree.
- a)1, 2 and 3
- b)1 and 3 only
- c)1 and 2 only
- d)2 and 3 only
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
What are the properties of a tree in a network graph?1. It consists of...
Tree: A tree is a subgraph of the main graph which connects all the nodes without forming a closed loop.
For a graph with ‘n’ nodes, the rank of tree = n – 1
Any tree for a given graph can be constructed with (n–1) branches.
Twig: The branch of a tree is called twig indicated by a thick line. Any tree with n nodes has (n–1) twigs.
Co-tree: The set of branches in a graph other than tree branches form a co tree.
Link: The branch of a co tree is called the link indicated by the dotted line. For any graph with n nodes and b branches, numbers of links = b – n + 1
Example:
A network with 4 nodes and corresponding graphical representation is represented below:
Now, if we remove the branches c, a, and f from the circuit, we will get the tree as shown below:
We observe that the tree for the given network contains all the four nodes of the network but does not form any closed path. (Statement 1 is correct, but Statement 3 is incorrect)
We can also observe that there are 3 twigs or branches of a tree in the given electric network. The number of nodes in the network is 4.
∴ The number of branches/twigs = Number of nodes - 1. (Statement 2 is correct)
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What are the properties of a tree in a network graph?1. It consists of...
Understanding Tree Properties in Network Graphs
In network graphs, trees serve as vital structures with specific properties. Let’s break down the properties mentioned in the question to understand why option 'C' (1 and 2 only) is correct.
Property 1: All Nodes Included
- A tree is defined as a connected acyclic graph.
- This means that it includes all nodes present in the original graph while ensuring connectivity among them.
Property 2: Branches Count
- A crucial characteristic of a tree is that if it contains N nodes, it will have exactly (N - 1) branches (or edges).
- This is because adding one more edge would create a cycle, violating the tree's acyclic nature.
Property 3: Closed Paths
- A tree cannot contain any closed paths (cycles).
- The essence of a tree is that every pair of nodes is connected by exactly one path, ensuring that no closed loop exists.
Conclusion on Options
- Since property 3 states that a tree has only one closed path, which contradicts the definition of a tree, this property is incorrect.
- Hence, the correct properties are 1 and 2 only.
Final Answer
- The correct answer is indeed option C (1 and 2 only), as properties 1 and 2 accurately describe the nature of trees in network graphs, while property 3 is false.
In network graphs, trees serve as vital structures with specific properties. Let’s break down the properties mentioned in the question to understand why option 'C' (1 and 2 only) is correct.
Property 1: All Nodes Included
- A tree is defined as a connected acyclic graph.
- This means that it includes all nodes present in the original graph while ensuring connectivity among them.
Property 2: Branches Count
- A crucial characteristic of a tree is that if it contains N nodes, it will have exactly (N - 1) branches (or edges).
- This is because adding one more edge would create a cycle, violating the tree's acyclic nature.
Property 3: Closed Paths
- A tree cannot contain any closed paths (cycles).
- The essence of a tree is that every pair of nodes is connected by exactly one path, ensuring that no closed loop exists.
Conclusion on Options
- Since property 3 states that a tree has only one closed path, which contradicts the definition of a tree, this property is incorrect.
- Hence, the correct properties are 1 and 2 only.
Final Answer
- The correct answer is indeed option C (1 and 2 only), as properties 1 and 2 accurately describe the nature of trees in network graphs, while property 3 is false.
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What are the properties of a tree in a network graph?1. It consists of all the nodes of the graph.2. If the graph has N number of nodes, the tree will have (N – 1) branches.3. There will be only one closed path in the tree.a)1, 2 and 3b)1 and 3 onlyc)1 and 2 onlyd)2 and 3 onlyCorrect answer is option 'C'. Can you explain this answer?
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