*The first thing to note is that, unless you are short on time, you should not be going to the tree that is next to the quest giver. This will only give you 1 Green Fallen Star and if it is just Greens you are looking for then you can obtain one of the other colours and trade that in for 2 Green Fallen Stars at the loot vendors.
The second is that your ability to obtain a specific colour of Fallen Star relies solely on the other home owners in each residential district. Since you can only have one per garden you will have to hunt about to find the tree that you hunger for like a wondering nomad. At least that would be the case if The Firm weren't SO FLIPPING AWESOME