I recently started getting strong enough to do ass to grass pistol squats. They are hard as fuck, and I can only a couple per side for now, but as soon as I get this hernia patched up, I will be back on the hard training. I don't see them as a prerequisite for tree climbers, but I find that the stronger I get all around, the easier all aspects of physical labor become, including climbing itself.
I was afraid that I would have a harder time hauling 20 extra pounds of meat around the tree, so I tried to stay slimmer, but I wasn't eating enough to get the fullest recovery from hard work, and I started struggling more and more with full work weeks. I weighed 165-170#. The pain got so bad, I went to a doctor of physical therapy, and learned that my piriformis was weak af, and the prescription was more work! I had to start doing excercises I had never considered doing, and then found more after our sessions that seemed to help a lot as well. I started doing deadlifts- current best before hernia was 185# for 16- and other traditional compound movements. I was up to 190# at the same belt size as at 165#, but have lost a few pounds waiting for surgery now, and I gotta tell you, moving myself around has never felt easier!
I highly recommend getting as jacked as you can while that is still relatively easy to do. PM me if you want a list of my favorite resources and stuff.