I usually do a few coatings of house latex paint as a primer before final painting (also using latex). I don't water it down and let it dry for at least a day before doing a second or third coat to try to minimize the warping that may happen to cardboard when exposed to wet paint. Latex is a polymer, which means once dry, it is like a thin layer of plastic. It is used for outdoor house painting due to its capability to withstand weather and UV exposure.
As long as you get all of the cardboard coated really well, chances are good that the prop will stand up to a season or two of periodic light rain/humidity. The key is to get it covered completely because even a few tiny spots where the cardboard is exposed is enough to act like a wick for moisture. And of course, if possible, bring the prop inside in the event of any storms/hard downpours, because even well protected cardboard may still wilt in hard rains/
You can also look into DRYLOK, which is a masonry waterproofer, and there is also a product called spar urethane which is a waterproofer for marine applications. Both of these options are going to be stronger than regular latex paint, but they are much more expensive and may be messy to deal with.
If you've already painted the tree then you'll need to check out clear coat sealers to protect the surface, but they'll likely be less waterproof-y than doing a layer of paint/drylok/urethane to start.