Orchard-planting problem

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 $t_{k}>{\frac {cn^{2}}{k^{3}}},$ where n is the number of points and tk is the number of k-point lines. Their construction contains some m-point lines, where m > k. One can also ask the question if these are not allowed.