TITLE: trees on six vertices
QUESTION [5 upvotes]: Show that there are exactly six non-isomorphic trees on six vertices.
("Introduction to Combinatorics" p.183)
The solutions were provided in the book.
But why can we be sure that there are ONLY six non-isomorphic trees?

REPLY [7 votes]: From Cayley's Tree Formula, we know there are precisely $6^4=1296$ labelled trees on $6$ vertices.
The $6$ non-isomorphic trees are listed below.

(These trees were generated as described in this answer.)
We also compute:

the size of the automorphism group $|\mathrm{Aut}(G)|$ of the tree $G$ (I did this using brute force checking of all $6!$ permutations) and
the number of labelled trees isomorphic to $G$, which is given by $\frac{6!}{|\mathrm{Aut}(G)|}$ by the Orbit-Stabiliser Theorem.

We see that $$6+120+360+90+360+360=1296=6^4$$ which matches Cayley's Tree Formula, so no tree is missing.