Any wire frame if left bare will have the ability to trip a GFI in wet weather.
You can insulate the wire part of the Mini Trees that will be used in the ground.
If you cannot do that, one other option is do not wrap the wire too tight around the wire frame and use a minimum of cable ties to hold them in place, do not overly tighten them either. What happens is the insulation is compromised in contact with the wire frame, One tight wrap might not trip a GFI but many will add up to a GRI trip in wet weather. Old (3 year or better) strings of lights insulation will not fair better in wet weather either, the UV protection is reduced and the insulation will start getting hard and getting micro cracks. If you see the insulation has turned a lighter green or white at the point of contact with the wire frame then you have a potential point of current leak to ground for a GFI trip.
Just my 2 cents worth.