So far I think
@sorce has the best idea for a beginner friendly setup, and you would need to worry about running out of water either.
If repeated flooding is an issue for any of your tree species, you could split the difference with the same general setup but the top part for the trees drains freely, and use the rotating sprayer heads for the drip type system you originally thought of. As a bonus there, if the trees are enclosed in a plastic bin, you'll only need a few or just one larger sprinkler to water every tree.
I'd start with the big plastic storage tubs, no lids necessary but you might get them anyway in case you have need of them in the future. Get 2 stackable ones of the same dimensions, except one two or three times as deep to be the reservoir. Cut/drill your drain holes in the shallow one, set it on top of your reservoir. A small pond pump would do it for moving the water. Get one rated for a pond the size of your reservoir, or something with an adjustable rate. Ask at your local garden or hydroponic shop, and they'll be able to help you. If you want to try the drip line sprayers, find one already made for a 1/4 inch line. Any decent plugin timer will do for turning your water on and off.
TEST IT THOROUGHLY BEFORE DEPENDING ON IT.
You could cut just a few drain holes and use some PVC to make fittings that allow for adjustable drain rates, but that'd take you a bit further down the diy rabbit hole. As would skipping the timer and in stead using a float valve.
I only mention these to point out how far you can go fairly easily. Projects like these can become an obsession of their own once you get going.
Enjoy!