I have some wood ones with that design that stay together just fine unless you pick them up wrong, so a little extra tension between pieces seems like it would help. (but yes, regular trees are good too)
Well, a die-cut is anything cut with a die. Dies come in different forms generally, but also for home/craft die cutting machines specifically. My machine is the Sizzix Big Shot. Sizzix makes other models and other companies make cutting machines. In my project the envelope, the window in the envelope, the insert card, and the trees and star are all cut with dies.
For the trees, I accidentally bought a knock-off die through AliExpress, not really understanding how that site works. So I thought I was buying a steel rule die which can cut heavier material such as chipboard - which the slot is obviously designed for. But what I got was something like a “Thinlit” die which can cut heavy cardstock, so I had to laminate layers of cardstock to get the trees to work. This is the one I thought I was getting.