A lot of things would happen, most of them bad.

Depending on what happened to the trees, you might lose your inputs of oxygen into the world. Other plants might pick up the slack, but you'd have a lot more free carbon dioxide around because trees normally store a lot of that stuff up.

Second, trees prevent a lot of erosion, and they alter the weather. Many parts of the world would become much more arid and desert-like as a result, even those parts that aren't otherwise covered with trees.

Last, because trees are something that evolution has favored, you can bet that what plants remained would eventually evolve to become new trees. This would take time, however.