"Tree-planting" puzzles are also known as "points and lines" puzzles. The English puzzle author and mathematician Henry Ernest Dudeney was very fond of them. In 1917, Dudeney published a collection of puzzles called "Amusements in Mathematics", which also contains the following classic puzzle:
Sir Isaac Newton's tree-planting puzzle: A farmer wants to plant 9 trees in such a way that there are 10 rows of 3 trees. How does he do this?
Note: the different rows are not allowed to be collinear.
P.S. I know of two different solutions. If you know any more that would be great and if you are able to prove that there aren't any more solutions that would be even greater. I consider two solutions different when there is no graph isomorphism possible between them.
Answer
A third solution. I've been searching systematically for them (I've eliminated all the combinatorial solutions other than ABC, A14, A25, A36, B15, B26, B34, C16, C24, C35 - it is the only one with geometric solutions).
To see that it is distinct from the others, note that it has a tree with 7 immediate neighbors.
I believe that is all of them, though. Once I double check to make sure I haven't missed something, I try to put something together to show it.
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