"Draw all non-isomorphic trees with 5 vertices."
I have searched the web and found many examples of the non-isomorphic trees with 5 vertices, but I can't figure out how they have come to their answer. How exactly do you find how many non-isomorphic trees there are and what they look like? Thanks for your time
Best Answer
One systematic approach is to go by the maximum degree of a vertex. Clearly the maximum degree of a vertex in a tree with $5$ vertices must be $2,3$, or $4$. If there is a vertex of degree $4$, the tree must be this one:
At the other extreme, if the maximum degree of any vertex is $2$, the tree must be the chain of $5$ vertices:
That leaves the case in which there is a vertex of degree $3$. In this case the fifth vertex must be attached to one of the leaves of this tree:
No matter to which leaf you attach it, you get a tree isomorphic to this one:
Thus, there are just three non-isomorphic trees with $5$ vertices.