Larry Barrett
Continuing the analysis from the note above, in the area of the orchard from 90-180 degrees from Joe's position I find straight lines that intersect two or more trees when the line y=mx has a slope m=9, 6, 9/2, 3, 9/4, 3/2, 3/4, 3/5 and of course when m=0 (the line Joe is standing on where he sees 7 trees in this direction).
Similar analysis for the area of orchard from 0-90 degrees, (here slope is negative with respect to Joe) when m=-12, -9, -6, -9/2, -3, -9/4, -3/2, -3/4 and of course when m=0 (the line Joe is standing on where he sees 6 trees in this direction).

So 9 lines in each area, including the baseline where Joe is standing.