Donald, you are nearly correct. There are 13 sites which are designated as group sites, the balance, are first-come, first-serve. There are three site which are closed due to nesting Osprey. And of course the host site. Which is where I am located.
There are also 2 group sites as well.
I believe that there are 35 with one dedicated to the camp host. You have to reserve a spot for camping up until Labor day. After Labor day it is first come first serve with no fees
According to the Forest Service website, 36. https://www.fs.usda.gov/recarea/boise/recarea/?recid=5094
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