Question
I havv a tennis court close to a forest.
what tree sap  would cause a nasty black residue on a tennis court?

Answer
Check and see if the sap is sticky and seems to fall is small mist like droplets. I would think that the trees have sucking insects (scale or aphids). These insects suck the plant juices fro the plant and secrete a substance called honeydew which is high in sugar content and sticky. A sooty mold will grow on this honeydew and give the black appearance. I would guess these are scale insects--you did not say the tree species around the court. These tree can be treated with an insecticide called Bayer Advanced Tree and Shrub Insect control to kill the sucking insects. This product is applied to the soil around the tree and the roots will carry the insecticide to the leaves and twigs and when the insects suck the plant juices they are killed. This treatment will last the full season and prevent further insect infestations. The sooty mold will go away after the sugar source is stopped. Check with your local garden type store for this product.
Here is a web link to the Bayer product.

http://www.bayeradvanced.com/tree-shrub-care/products/12-month-tree-shrub-insect...