Drawing all non-isomorphic trees with $n = 5$ vertices.

Question

I followed the instructions in this question to create the trees for n = 1, n = 2, n = 3, n = 4, and n = 5. Note that I created rooted trees instead of free trees because it was easier. You can get the free tree versions by just messing with the branch arrangements. The problem is taken from my book, which asks me to draw all non-isomorphic trees with $5$ vertices. Here's what I did:

The book has the following answer for $n = 5$. Two of my trees (the two on the left labeled $n = 5$) match the ones given by the book, but the third tree I drew does not match the tree they have where one node has 4 branches.

What exactly did I do wrong?

Answer 1

Your last two five vertex trees are isomorphic. Each has one vertex with degree $3$ and that vertex has chains of $1,1,2$ leaving it. The three leaves of the first $n=4$ graph are equivalent and you added a leaf to two of them. You need to add a leaf to the degree $3$ vertex to get the last $n=5$ tree.