Orchard Planting Problem/Classic Form

Classic Problem
A given number $n$ of trees are to be planted in an orchard so as to make the number of rows of $3$ trees the largest number possible.

That is:

$n$ points are to be configured in the plane so that the number of straight lines that can be drawn through exactly $3$ of these points is maximised.

Solution
Let the maximum number of rows of $3$ for a given number of trees $n$ be denoted $t_3 \left({n}\right)$.

Then we have: