Okay there are some super in depth answers to this and as others have stated we don’t really NEED trees as much as one would think…but this is ELI5 so here is a fairly reasonable thing to think about even if we did. Winter for you is Summer for the opposite hemisphere. So as your trees fall, others are growing back. Kind of like the balloon effect. squeeze one side (Winter) and it gets smaller but the other side (Summer) gets bigger. Then flip it and it works the other way around.
The other things posted here are true, but the big reason is that winter doesn’t last long enough for us to meaningfully deplete the oxygen in the air. If you look at carbon dioxide, of which there is much less, you do indeed see a [very noticeable seasonal cycle](https://en.wikipedia.org/wiki/Keeling_Curve#/media/File:Mauna_Loa_CO2_monthly_mean_concentration.svg) where CO2 levels peak in the winter of the northern hemisphere (where most land is, and thus most land plants live).
This seasonality happens worldwide, though, because the atmosphere effectively mixes gases from every part of the world to every other part over timescales of weeks to months.
The oxygen in the air isn’t really there because of the current trees, or any other current organism making oxygen via photosynthesis. Even if all the trees, algae, and bacteria that perform photosynthesis stopped suddenly, it would take a while to deplete the levels noticably. There’s a massive surplus of oxygen in the atmosphere. The larger northern hemisphere having a portion of its deciduous plants stop for a portion of a year winter is nothing.
How did the oxygen surplus get there? The oxygen is there from a couple billions years of photosynthesis, and then the organic matter not decaying or being burned. See fossil fuels, and what we are burning is just a tiny fraction of the organic molecules trapped that never turned back. Lot of carbon trapped in the ground, that never reunited back with the oxygen in the air.
Trees do not produce the majority of the oxygen. 50-80% of all oxygen comes from the sea where it is mostly plankton that produces it. so 20-50% is land and there is lot of other plants than trees.
But even if they did you would not see a dramatic change because the amount of oxygen that is used is not very high compared to the amount in the atmosphere.
If no oxygen was produced and the amount of animal was like it is today (do not as ask me what they eat if there are no pants) the all oxygen in the atmosphere would be used up in 52 000 years. That also assumes that all oxygen can be used.
Let’s say winter is half a year so 1/100 000 of the oxygen would be used by animals
That is not the whole story of oxygen usage because it is used when nonanimals like fungus, bacteria break down biological matter.
So more oxygen is used than just by animals but those processes then get slower in the winter especially if it is below freezing.
The result is that there is a minimal change in oxygen level in winter vs summer we talk about 0.02 to 0.03 percent. I’m not sure the number is the percentage of the atmosphere or percentage of the average oxygen level but regardless it is not a lot.
There is an extreme amount of oxygen in the atmosphere. We have enough oxygen in the atmosphere to last for maybe a thousand years of winter. So we do not notice much of a difference in the oxygen levels between summer and winter, even though there is a tiny bit. What we can measure this better is in the carbon dioxide levels. There is more landmass in the northern hemisphere then in the southern hemisphere. So there is more trees and vegetation in the northern hemisphere then the southern. This means that more carbon dioxide is converted into oxygen when it is summer in the north then when there is summer in the south. And since there is not that much carbon dioxide in the atmosphere this does make out a noticeable change in levels. The difference is about 5ppm. That used to be a greater then 2% change in carbon dioxide levels but is now fast approaching only 1% due to the increasing levels of carbon dioxide in the atmosphere.
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