  Consider a set of $ n $ points on a plane. A line containing exactly $ 3 $
out of the $ n $ points is called a $ 3 $-rich line. The classical orchard
problem asks for a configuration of the $ n $ points on the plane that
maximizes the number of $ 3 $-rich lines. In this note, using the group law in
elliptic curves over finite fields, we exhibit several (infinitely many) group
models for orchards wherein the number of $ 3 $-rich lines agrees with the
expected number given by Green-Tao (or, Burr, Gr\"unbaum and Sloane) formula
for the maximum number of lines. We also show, using elliptic curves over
finite fields, that there exist infinitely many point-line configurations with
the number of $ 3 $-rich lines exceeding the expected number given by Green-Tao
formula by two, and this is the only other optimal possibility besides the case
when the number of $ 3 $-rich lines agrees with the Green-Tao formula.
