I just attempted to complete the "Chop down some trees" goal on my priest by planting trees and immediately chopping them down with a hatchet.
This didn't seem to trigger the goal, which is unfortunate as it means I've wasted about 1 point of Faith on it.
If the goal requires a felled tree to be spawned, the description should probably be changed, and if not, something is wrong with the goal itself triggering.