In discrete geometry, the original orchard-planting problem (or the tree-planting problem) asks for the maximum number of 3-point lines attainable by a configuration of a specific number of points in the plane. There are also investigations into how many k-point lines there can be. Hallard T. Croft and Paul Erdős proved

$${\displaystyle t_{k}>{\frac {cn^{2)){k^{3))},}$$

Integer sequence

Define ${\displaystyle t_{3}^{\text{orchard))(n)}$ to be the maximum number of 3-point lines attainable with a configuration of n points. For an arbitrary number of n points, ${\displaystyle t_{3}^{\text{orchard))(n)}$ was shown to be ${\displaystyle {\tfrac {1}{6))n^{2}-O(n)}$ in 1974.

The first few values of ${\displaystyle t_{3}^{\text{orchard))(n)}$ are given in the following table (sequence A003035 in the OEIS).

n	4	5	6	7	8	9	10	11	12	13	14
${\displaystyle t_{3}^{\text{orchard))(n)}$	1	2	4	6	7	10	12	16	19	22	26

Upper and lower bounds

Since no two lines may share two distinct points, a trivial upper-bound for the number of 3-point lines determined by n points is

$${\displaystyle \left\lfloor {\binom {n}{2)){\Big /}{\binom {3}{2))\right\rfloor =\left\lfloor {\frac {n^{2}-n}{6))\right\rfloor .}$$

${\displaystyle {\tfrac {6n}{13))}$

$${\displaystyle \left\lfloor {\frac ((\binom {n}{2))-{\frac {6n}{13))}{3))\right\rfloor =\left\lfloor {\frac {n^{2)){6))-{\frac {25n}{78))\right\rfloor .}$$

Lower bounds for ${\displaystyle t_{3}^{\text{orchard))(n)}$ are given by constructions for sets of points with many 3-point lines. The earliest quadratic lower bound of ${\displaystyle \approx {\tfrac {1}{8))n^{2))$ was given by Sylvester, who placed n points on the cubic curve y = x³. This was improved to ${\displaystyle {\tfrac {1}{6))n^{2}-{\tfrac {1}{2))n+1}$ in 1974 by Burr, Grünbaum, and Sloane (1974), using a construction based on Weierstrass's elliptic functions. An elementary construction using hypocycloids was found by Füredi & Palásti (1984) achieving the same lower bound.

In September 2013, Ben Green and Terence Tao published a paper in which they prove that for all point sets of sufficient size, n > n₀, there are at most

$${\displaystyle {\frac {n(n-3)}{6))+1={\frac {1}{6))n^{2}-{\frac {1}{2))n+1}$$

${\displaystyle t_{3}^{\text{orchard))(n)}$

This is slightly better than the bound that would directly follow from their tight lower bound of ${\displaystyle {\tfrac {n}{2))}$ for the number of 2-point lines: ${\displaystyle {\tfrac {n(n-2)}{6)),}$ proved in the same paper and solving a 1951 problem posed independently by Gabriel Andrew Dirac and Theodore Motzkin.

Orchard-planting problem has also been considered over finite fields. In this version of the problem, the n points lie in a projective plane defined over a finite field.(Padmanabhan & Shukla 2020).