The simple answer is, no, you can't. These trees are permanent fixtures of the town and cannot be tapped or chopped. However, this only applies to the trees in the center of town. All the trees in the Cindersap Forest, the mountains, the back lane behind your farm, and near the Bus Stop are all up for grabs and can be tapped as well as cut down. Then when you cut them down, they grow back fast, with a 60% chance of jumping straight to growth stage three at the end of every day after you chop them down.