How to calculate sizing for treehouse footing?
Hmmm...no digging in the ground and no concrete. 

Sounds like the four perimeter beams will be your foundation. They’ll need to be treated for ground contact (pressure treated) as you indicate. 

Your total load will be about 220 lbs. (Live Load) as you indicate, plus about 800 lbs. in lumber and material (Dead Load). I did not add snow load as I’d assume if there’s snow, then there’d be no “kid” load. 

If you don’t remove the topsoil, your soil bearing value will be about 200 - 400 lbs. per square foot. 

Assuming a total load of about 1,020 lbs. distributed at the perimeter: 13’ x 13’ x 4” wide beam for a footer allows a total footprint of about 17 square feet. 

So, with a total load of 1,020 lbs. / 17 sq. Ft. = a soil bearing value of 60 lbs. per square foot for 4x beam footings. 

Option 1: Use pre-cast square concrete footings at the corners. So, 4 - 16” x 16” footings will give you a footprint of about 7 square feet and a soil bearing value of about 145 lbs. per square foot, which is also very reasonable.