Certified GKH Gardening Expert
The only animal that I could think that might do that would be deer or a human. How tall is the tree? And when the branches are removed, are they clean cut or gnawed?
Insects normally will attack the leaves and flowers of the tree, not the stems of the new shoots. I suspect that it is a rodent of some kind. The young shoots would be tasty to them and they may be trying to "trim" them to haul them away, but sometimes the branches are too heavy and they fall to the ground instead. Squirrels and chipmunks are known to do this, but not often. Normally only of they can't find other food, so it would be an early spring thing, not this late in the year normally.
Your best option to really discover the pest is to place some sticky traps, or even reversed duct tape around the trunk and at a few places on the tree. These should trap at least a few of the bugs, if they are bugs or maybe the rodent in question.