I have no experience with climbing or its related knots. Would this just be a back up connecting to one side of the saddle as a fail safe? Thanks!
Yes. This only backs up your "Rope Grab" device on your tether. Say you had a hand on your prussik or ropeman and you slipped...you might go for a ride. An Alpine butterfly tied on your tether attached to your belt loop would prevent.
If your tether breaks you hit the ground. Likely your tether is 11-13mm Static Kernmantle rope rated around 6-7000lbs. You probably have Figure 8 knot that you pass through to girth hitch the tree, so that decreases the rope strength by 20-40%...so your tether is good for say 4000 lbs. If your body experiences a 4000lb force...well your guts turn to jelly.
If you want a "true" secondary back up keep your lineman's belt attached to yourself and the tree and run a prussik hitch attached between both sides of the tight lb.
Climbing Geek Out Notice: When working at heights you should have a Primary and Secondary. Primary is whatever your weight is resting on. That can be your hands and feet, your lineman's belt or whatever you are using to "work" rope.
The secondary is your back up in case primary fails. No weight should ever be in the secondary.
In the case of saddle hunting we don't have a true secondary because our weight is on our teacher and we have no back up.
Be advised I'm not an arborist and only work /train other guys to work on towers. So my tree experience is limited but the principles are basically the same.