I had the same problem with a plum tree, although a smaller infestation.


One natural/non-Chemical treatment is diatomaceous earth mixed with some dish soap and sprayed on them and the area. The DE and soap will cut into their exoskeletons and dry them out. Let me know if you want some DE and are going to Albion. I'll give you a small bag. A few treatments will get at the, 

You can also use a termicide or insecticide from the hardware store.

With either treatment, cut away and destroy (burn) the infected wood and any dead wood (sounds like you took out the whole tree) so they don't spread to other trees or your home.

Note termites live in huge colonies so, if they were at the bottom, it's possible they're throughout the rootbed and ground. Call exterminator.

My opinion is different than you've gotten so far. The bay area is completely infested with termites and the probability of you being able to exterminate all the ones in your yard is probably pretty low.

The ones you found in your yard are subterranean,  They hatch and fly to find new territory so even if you get the ones in your yard you'll have more from surrounding areas very soon.

Best bet is to get an inspection of your house then take any steps recommended to exterminate near your house and also do whatever suggested to keep your house safe and leave the rest alone.

Termites will not eat live wood, but love dying trees and their root systems.