You do not get black dots from trees. The material that causes the stain on cars, tents, etc is the results of natural sugars (honeydew)that result when aphids feed on the leaves of various trees.
The aphids can digest all parts of plant juices except the sugars that are present in abundance. These sugars are expelled and fall to the ground as waste.
Fungus feeds on the sugars causing the dark spots. Anything that you place on your tent to remove the spots may very well effect the water proofing of the material.