Question:
Ignoring vertex labels, how many distinct trees are there with 5 vertices? Draw each such tree, and justify your conclusion that there are no more. (You may assume without proof that any tree on n vertices has exactly n-1 edges.)
Where I am at so far
I understand that a tree is a graph which is connected (which means from any vertex you can walk to any other vertex) and acyclic (which means the graph contains no cycles).
The only distinct trees on 5 vertices I can think of is a star with 5 vertices and a path of length 4, denoted $P_4$. Are there any other distinct graphs? Also how would I prove that there are no more?
Build your tree step by step:
1- A 5-vertex tree must have a vertex: 0
2- This vertex must have a neighbour (a tree is connected): 0-0 (no other option)
3- At least one of these vertices must have another neighbour: 0-0-0 (again no other options).
4- Adding a fourth vertex you have two options (which are?).
5- Then in both cases you have again exactly two options to add a fifth vertex. This yields $4$ trees, but two of them are actually identical.
In the end you have the two trees that you found, plus the Dynkin diagram $D_5$.