Most answers are literally missing the forest through the trees (by focusing on one and only one solution), and ignoring the original problem.
The OP asked two questions:
As a result, I need a lot of wood and my supply is running low.
and steered people towards one solution:
What can I do differently to make my tree farm more efficient?
Since no one has answered the original question -- I'll present the two solutions to get a lot of wood:
The fast way
Ironically, I found the fastest way to get a lot of wood was not to have a tree farm at all! I found it far more efficient (time wise) to just create a new world, pick a direction say East, and just chop every tree until you hit the ocean. Use the mirror (or Save & Exit then reload the world) and travel the opposite direction as a lumberjack. YMMV but on a small world travelling in just one direction netted:
- 1015 Wood
- 627 Rich Mahogany
- 158 Palm wood
- 102 acorns
Travelling in both directions netted a grand total of:
- 1459 wood
- 748 Boreal Wood
- 627 Rich Mahogany
- 255 Palm wood
- 26 Ebonwood
- 223 Acorns
And the stats:
- Total wood: 3115 / Total time: 19 mins = ~ 163 wood/min
- Total platforms: 6230 platforms = ~ 327 platforms/min
Rinse and Repeat. This should be enough wood to make those extra long sky bridges arenas. The bonus is that you can save all those acorns to (cough) seed your tree farm. :-)
The slow way
Traditionally, the tree farm is most popular -- partially due to aesthetics -- but it is dog slow.
First, you have to prepare a flat surface.
I recommend 1,000 tiles of dirt (optionally also sand if you want/need Palm wood. If you want jungle trees your tree farm needs mud deep Underground)
- Plant trees with 4 spaces between saplings. This gives 2 block radius on either side of each tree.
- Use a hammer so you don't have dirt wallpaper behind as this will block the trees from growing.
- Place torches below ground so as not to impede the growth of the trees
Maintenance of the tree farm will require you to continually clear all the grass. If you use a hammer and aim in the sky you can clear all the grass and leave the trees & dirt safely alone.
To speed up growing the tree farm, hole up like a hobbit near the surface, preferably with a lava trap above, and leave the world running over night. In the morning you should have most of the trees growing and you'll have also made a few extra coins.
Once you have a large enough tree farm setup it will produce more wood quantity-wise in the long run but setup & maintenance time is just a huge PITA compared to just clear cutting on a brand new temporary world.