the green arrow will always appear when you place every flower, regardless of whether or not the solution is correct.
the goal is for you to deduce the rules of planting the flowers, since the game will not tell you these rules. in each garden there is exactly one correct arrangement of flowers. you'll know that a garden is correct if the rules you have deduced only allow one possible arrangement of flowers. if it seems there are no possible arrangements, or more than one, you have an incorrect or incomplete understanding of the rules.
you can also receive hints in-game, if you click "next chapter" while in a set of gardens.