If you're amenable and able to use mods, TreeCapitator and Timber! both make all the wood of a tree drop as soon as its bottom is cut.
The speed-up in harvesting is immense.
In vanilla Minecraft it's not possible to automate tree farms.
The only way to speed harvesting of wood is to use faster tools, which means diamond or gold axes, and/or the Efficiency enchantment.
If you're one of those players who has little use for powered rails and therefore little use for gold once you have one clock, throwing it into fast-but-fragile gold tools is a good use for it.
gaming.stackexchange.com/users/4163/SevenSidedDie
When harvesting wood in vanilla minecraft, I follow this process: Use a level 1 enchantment on an iron axe.
It will almost always be efficiency I. Get a stack of bone meal, and several BIRCH saplings.
Find an open and relatively flat area.
Plant a sapling, then use bone meal.
Harvest the wood blocks from the tree.
Walk 3-4 spaces, plant another sapling, use bone meal, then chop down.
Repeat until you run out of bone meal.
Using this method, you don't have to worry about leaf blocks (because of decay), you will never have to climb trees (because birch trees don't grow higher than 7-8 blocks), and you can repeat it as often as you'd like.
You can get several stacks of wood in a matter of minutes using this method.
gaming.stackexchange.com/users/27903/Gigazelle
If you are still punching trees, create and use an axe (as even a stone one is faster than the fist), then you can enchant it with Efficiency for extra speed (and Unbreaking will make it last longer).
gaming.stackexchange.com/users/13414/ratchet freak
I cut down on harvesting time by reducing: walking leaf harvesting climbing (to none)
I plant my trees in a block with no spaces.
For example: I have an area of trees 7 x 5 where I have 30 saplings planted.
The 5 spaces that don't have saplings have a torch - this is so each sapling always has a light source.
There are also torches around the perimeter - each sapling receives light from an adjacent, non-diagonal space.
Currently I use oak saplings, but in order to restrict the growth (and thus be able to reach the highest wood block from the ground), I have a "lid" of cobblestone
(there are 6 spaces between the dirt and the cobblestone).
I've tried this without the lid, but it just turns into a massive tree, that is more trouble to harvest than a field of regular trees.
Russell Whitchurch mentions birch trees could be used to restrict the height as well.
Using this method with the farm listed above, a single harvest (when most of the saplings have grown) yields more than 2 stacks of wood, and enough saplings to replant.
gaming.stackexchange.com/users/30989/ChargingPun
You have a few options:
First is to use better tools.
Of course, this requires resources, but the investment will significantly speed up your harvest rate.
A diamond axe with Efficiency V and Unbreaking III will both last a long time and harvest very quickly.
Second is to optimize your harvesting area.
This can involve: Creating a Haste II beacon to speed up gather times.
Flattening the land to allow for faster travel.
Creating a roof to limit the growth height of trees (this will slow growth)
so you don't have to leave the ground to completely access all the wood.
Plant trees in an easily navigable pattern, such as a grid.
Use some system to mark where trees are for faster replanting.
Third is to create a semi automatic tree farm design.
This uses pistons to push newly grown trees into a solid block of wood, greatly reducing travel times and allowing for much faster harvesting.
This can be seen here:
Finally, you can create a fully automatic tree farm by abusing the Wither to break blocks, as shown here:
Be aware that the last two methods require a great deal of effort, but will provide long term, sustainable wood income.
The first is best for short term solutions, unless you are fairly liberal with diamonds.
The second is perhaps the most common method, and will greatly speed up your harvesting permanently, but requires more attention than later methods.
Certain techniques from the second method (such as the Haste beacon) are also applicable to other methods (such as the third).
gaming.stackexchange.com/users/65586/Aze
I seem to spend a lot of time harvesting trees.  What can I do to reduce the amount of time I spend harvesting wood?
How can I speed up harvesting trees?
gaming.stackexchange.com
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gaming.stackexchange.com/questions/105654
gaming.stackexchange.com/users/30989/ChargingPun
You can enchant your axe for efficiency or use a mod like Treecapitator and Timber! While there are way, up to a point, to harvest trees faster, mod usage is a common option too.
Mod usage is adviced. The community proposes few things that might help speeding up the harvesting.