a tree is a connected undirected graph with no cycles, how many non isomorphic trees with 5 vertices exist? 

A tree is a connected, acyclic graph, and when considering **non-isomorphic trees**, we're looking for distinct tree structures that cannot be transformed into one another by renaming vertices.

To find the number of non-isomorphic trees with 5 vertices, we can either systematically list the possibilities or use known results for small numbers of vertices. For **n = 5**, the number of non-isomorphic trees is already known:

There are **three** non-isomorphic trees with 5 vertices.

Here are the different non-isomorphic trees for 5 vertices:
1. A star graph \( K_{1,4} \): One central vertex connected to four others.
2. A path graph \( P_5 \): A single line of 5 vertices connected in sequence.
3. A tree with two vertices of degree 2: Two paths that join in the middle, i.e., two vertices of degree 2 connected by one vertex of degree 3.

These represent all distinct tree structures possible with 5 vertices.

Would you like further details on these tree structures or how to verify their isomorphism?

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Here are some related questions:
1. What are the properties of a tree graph?
2. How can you verify that two graphs are isomorphic?
3. How many non-isomorphic trees exist with 6 vertices?
4. How does Cayley's formula help in counting labeled trees?
5. What is the significance of degree sequences in classifying trees?

**Tip:** In graph theory, non-isomorphic graphs cannot be mapped onto each other by simply renaming vertices, as their structures differ.

A tree is a connected undirected graph with no cycles, how many non-isomorphic trees with 5 vertices exist?

Explore the number of non-isomorphic trees with 5 vertices. This problem discusses three distinct tree structures: the star graph (K₁,₄), path graph (P₅), and a tree with two vertices of degree 2. Understand tree isomorphism and distinct tree structures, ideal for students studying graph theory at the college level.

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