Your backyard is a protected area, right? In your backyard, massive deforestation can never happen, since it is your backyard, nobody but you can cut down the trees there.

Nowadays government owns these "protected areas" and/or prohibits deforestation in them - portions of the Amazon rain forest, for instance.

I am reading the Wikipedia article, and it does not seem like government is doing a good job protecting Amazon rainforest. In fact, seems like they have significantly contributed into destroying it. Which makes sense, because they don't care about the forest, they only care about getting elected.

In order to find out who might be able to protect the forest better, we need to look at who the current legitimate owners of the rainforest are. In order to become a land owner, it is not enough to point your finger at the entire forest, and say "this is mine!", like governments do. You need to either buy it from the previous owner, or you need to be actively using it. Wikipedia mentions: indigenous territories continue to be destroyed by deforestation ... the Peruvian Amazon indigenous peoples' rainforest communities continue to disappear ... the Urarina continue to struggle to fight for their cultural survival and the fate of their forested territories. So, looks like the legitimate owners are the indigenous people. And whomever is cutting down the trees in their territories, is essentially stealing their land. This is not a nice thing to do, I, as an anarcho-capitalist, would not support that.

Now, buying land is OK, so we have two options here. Either the indigenous people are ready to sell that land for cheap, or it is expensive.

If it is cheap, environmentalists can easily buy large portions of it, then hire rangers to patrol and protect it.

If that land is expensive, massive deforestation would be expensive, too. It would be cheaper for the loggers and farmers to buy smaller plots of land, and resort to more sustainable techniques which do not require buying new plots of land every year.

So, private property could be an answer to the rainforest issue, but Brazilian government has already proved that it cannot be the answer.