Presumably you have a tree farm to harvest as much wood as you can get. The canopy in your screenshot is hanging around because there's still good wood in there! Go and harvest it, and the leaves will decay on their own.
Branching trees are most efficiently harvest top-down, unlike simple trees. Get up on top of that tree, then mine downwards toward the visible blocks of wood. Just remember the cardinal rule: never mine the block you're standing on without knowing it's safe! Don't worry about catching the log resources that fall—get those once you're safely back on the ground.
Once you've taken care of the branches, harvest the trunk from top to bottom. That will get you back to the ground with the least danger. In the case of this sad, trunkless specimen, give it a prosthetic "trunk" made out of dirt or gravel before you get up into the leaves, so that you will have a safe stack of blocks through which to mine your way down to safety.
A nice side-effect of harvesting trees this way is that, by the time you're back on the ground, it will have dropped plenty of saplings. No more running to your storage chest or waiting around empty-handed waiting for a sapling to drop just to replant one tree!
Even tree farms with roofs will generate branching trees occasionally. You might as well keep your nice outdoors tree farm, but upgrade it with a perimeter catwalk and stairs/ladders to give you quick access to the upper surface of tree canopies when your farm pops out a branching tree. (Also, running around on top of trees is inexplicably fun.)