Generally, no, and I live near a very tree-dense state park, although it does have a normal road that cuts through it. The rule of thumb is let the Garmin come on completely, go to the main computer screen showing the speed, time, etc. and you should be synched to satellites by then. This process takes longer when you're in an area where you've never ridden.

The improvement that occurs once you leave the trees could be due to either the absence of the trees or the fact that your unit has locked on to more satellites by that time.