We don't recommend using a power washer on tree trunk; it's hard to predict with certainty, especially given the varying horsepower of the motors of different pressure washer units, but it may risk blasting off some bark layers. Instead, you can just give sooty mold time to weather off on its own. Once the source of sugars (in this case sap, in other cases honeydew from sap-sucking insects) is diluted or has subsided, the sooty mold fungus will run out of food and die off.

Another option: some horticultural oil products may include instructions for using it to remove sooty mold, though that is not generally its primary function. We don't know how long it would take a treatment to produce any visible results, but that approach should also cause it to eventually wear off.

Miri