There are also biological routes to making cis-isoprene which involve microorganisms instead of plants or trees. None of them are commercial yet as far as I know.
They "harvest" the rubber trees at the end of their service life and make cheap furniture from them, but as noted the rubber is produced by tapping live trees and collecting the latex.
I was surprised to read on Wikipedia that natural rubber production is still about 40% of total rubber production, so I did some digging and the numbers are more or less accurate. I would have thought that SBR would have dwarfed it completely, but there you go- it's still cheap enough and the properties are great for some applications.
Where we need a bit of help is with the disposal of the used tires. They burn cleanly when burned in a properly designed combustor and have an incredibly high energy content, and are far cleaner as a fuel source than most coals. But because everybody has seen at least TV images of smoky, sooty burning tires (which is more pyrolysis than combustion), people think they are a dirty fuel and won't permit them to be burned in many jurisdictions. So instead, they sit in piles and burn by accident, creating major environmental impact even after the firefighters manage to put them out which is not easy by any stretch of the imagination. There are some re-use options (such as the use of recycled rubber crumb in asphalt etc.), and some people have tried to pyrolyze them to make liquid fuels, but there are still mountains of used tires to be found.