I wouldn’t worry to hard about it. I think everyone who has driven a truck has hit some branches. A couple years ago I was traveling on a 2 lane state road through a small town with plenty of trees over the road. It was early summer so the trees had plenty of leaves and were maybe sagging more than usual and I was hauling an excavator backwards of how I usually do so the boom was to the front. Unfortunately it contacted a rather large branch that pulled down and ended up breaking off. Unfortunately a car happened to be coming towards me at the time and hit the branch right after it hit the road. No damage to my truck and a broken hydraulic hose on the excavator but it made for a long afternoon and took a while to get sorted out because somehow it was my fault…it ended with the town paying for the damage to the car.
I've Stopped, opened the hood, stood on the front tire, and cut a branch off before. You don't have to just ram into things. Looking up is also a part of the job.