No. If you chop down a tree, you will be left with a stump, and it will not regrow. Stumps can be cleared using the Shovel. Trees will be chopped down if you hit them 3 times with an Axe in a single day (this will not happen if you use the Flimsy Axe or the Stone Axe). However, Wood is replenished every day, so you can use trees as a regular source of materials. Also, you can grow more trees by planting saplings.