> Is there math behind this? Even back-of-napkin math?

Here. Maybe this qualifies for "back of the napkin" [0].

Quoting:

"Changes to our atmosphere associated with reactive gases (gases that undergo chemical reactions) like ozone and ozone-forming chemicals like nitrous oxides, are relatively short-lived. Carbon dioxide is a different animal, however. Once it’s added to the atmosphere, it hangs around, for a long time: between 300 to 1,000 years. Thus, as humans change the atmosphere by emitting carbon dioxide, those changes will endure on the timescale of many human lives."

It isn't a matter of just looking at a tree. You have to consider the entire process. This includes energy and time. The time scale for CO2 in the atmosphere isn't measured in days or years, it's measured in centuries or thousands of years.

You can't recapture the CO2 produced by burning a thousand trees in a few days on the same time scale. The real recapture, when all things are considered, is orders of magnitude greater in time. And, of course, because everything in real life is a complex multivariate problem, there are a million other factors that make the process far less from the idealized image a lot of people have in their minds. A lot of those variables are unknown to us. My guess is a good deal of it isn't even measurable. We are still learning a lot. For example, again, from the article, quoting:

"For as long as we can remember, we’ve talked about Earth’s tropical rainforests as the ‘lungs’ of our planet,” he said. “Most scientists considered them to be the principal absorber and storage place of carbon dioxide in the Earth system, with Earth’s northern boreal forests playing a secondary role. But that’s not what’s being borne out by our data. We’re seeing that Earth’s tropical regions are a net source of carbon dioxide to the atmosphere, at least since 2009. This changes our understanding of things."

Going back to the back of the napkin calculations. A tree absorbs about 25 kg of CO2 per year.

A tree.

Not a seedling.

Not a twig.

A seed just weeks out of germination isn't going to absorb 25 kg of CO2 in a year. It takes years, maybe decades (likely species dependent) for it to become a tree that can capture carbon at that rate. Some trees never reach that rate of capture. They are not large enough.

And so, you don't burn a tree in a day --one that captured thousands of kilograms of CO2-- plant a seed and instantly start capturing 25 kg per year. It doesn't work that way.

This is why including variables such as time, growth rate, species and many more is important. Without them we are constructing a fantasy. It might take twenty years for a tree to reach this 25 kg/year absorption rate. By that time you have twenty years of forest fires that have contributed CO2 to the atmosphere that will "stick" to it at a time scale longer than human life or even human generations.

My guess is that the shape of the rate of CO2 absorption for any plant or tree is roughly a sigmoid function [1]. I tried to look this up but Google isn't being helpful in this regard. All you can find are numbers for mature trees. I want to know the rate of absorption per year from seed to maturity. Beyond that, I want to know how this behaves after a tree reaches maturity. Does it continue at the same rate? I have a sense this might not be the case. This is why I am guessing it must look somewhat like a sigmoid, with the possible exception that, once a certain age is reached, it might not stay flat but it might actually decline to some lower rate of absorption. This, in particular, makes sense to me in the context of a canopy or a forest. Interaction between trees and other factors means sunlight can only penetrate to a certain depth into the canopy. This likely means the rate of absorption has an upper bound or might experience a reduction, maybe even a cyclical reduction.

And then there's NASA's discovery that the tropical rainforests have actually become net CO2 contributors. I don't fully understand that yet. It raises a lot more questions in need for answers.

It is all too easy to fall into what I like to jokingly call the "assume a cow is a uniform sphere of milk, one meter in diameter" syndrome. Anyone who has studied physics to a reasonable depth is familiar with the many "assume..." simplifications we are forced to use while learning. Reality is too complex and it would be very difficult to understand things we we stopped to account for all variables while learning. That does not mean that we can then take the cow assumption and apply it in real life to every problem. Sure, a tree can absorb a lot of CO2...but this isn't some magical process that happens instantly the minute you are done burning it. Not even close. A cow is not a uniform sphere of milk. It just isn't.

We have much to learn.

[0] https://climate.nasa.gov/news/2915/the-atmosphere-getting-a-...

[1] https://en.wikipedia.org/wiki/Sigmoid_function