Brute force, and nondeterministic

    NestWhile[l \[Function] 
     If[
        FreeQ[VertexDegree@#, 2] \[And]
        TreeGraphQ@# \[And] 
        And @@ Table[\[Not] IsomorphicGraphQ[G, #], {G, l}]
      ,l~Append~#,l
     ] &@RandomGraph@{10, 9}
    , {}, Length@# < 10 &]
With `\[And]`, `\[Not]` and `\[Function]` counted as one character, I count 149 characters when removing the unnecessary whitespace (all whitespaces).

I swear it worked immediately on my computer :)

[![all graphs][1]][1]

Test with `Or @@ IsomorphicGraphQ @@@ Subsets[%, {2}]`

  [1]: https://i.sstatic.net/obZpK.png