Applied combinatorics: graph theory non-isomorphic trees help please?

Hi I wonder if someone can help me with the following questions:

(a) List all non-isomorphic trees (not rooted) on 12 vertices with each vertex of degree either 1 or 3.

I get only one such tree with 5 internal vertices of degree 3 and 7 leaves (of degree 1) I think this is correct but if I'm missing something please let me know.

The question I'm having trouble with is this one:

(b)Explain why your list in (a) contains no repetition (i.e. why the trees are all non-isomorphic), as well as why your list contains all such trees.

I used trial and error to find the tree so how do explain that it is unique?

Any help would be greatly appreciated.

Thanks in advance.