As of now I have 4 assuming I got this right. I apologize for formatting because I have no idea how I'd display this in text but here I go. a b c d are my points. - means there is a connection between them
I have
a - b, b - c, c - d
a - b, b - c,b - d
a - b, a - c, b - d
a -b, a - c, a - d
Hope this makes sense.
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$\begingroup$There are actually just two, and you’ve found each of them twice.
Your first and third trees are isomorphic: the third is c--a--b--d and is thus a single chain, like the first. Similarly, your second and fourth are isomorphic: each has a single vertex of degree $3$ connected to three vertices of degree $1$.
Once you have a vertex of degree $2$, like this:
*----*----*there really are only two ways to proceed. You can attach the fourth vertex to the vertex of degree $2$ to get your second and fourth trees, or you can attach it to one of the vertices of degree $1$ to get your first and third trees.
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