I don't think ladies, or even gentlemen, were disappearing behind trees much - 
they didn't need to. 
I've been waiting to someone to mention:-
1. The chamber pot, a part of every household before indoor sanitation,
2. The earth closet which was the standard outside facility before sewers, 
both in towns and the countryside, effective, not smelly and easy to make if 
you know how. It would not be difficult to construct additonal closets for a 
planned event. They could be closed over or emptied later.
Far from being a problem an influx of people was a potential benefit. In Europe 
it was common to collect human waste (liquid and solid) as manure or for 
tanneries. I have seen at least one record which explained the poor soil in a 
parish was becasue it was a long way from any town, and couldn't get hold of 
it!