Stick eating

Question
Our 2 year old chocolate lab that we just rescued loves to eat sticks outside. If he doesn't find one on the ground, he'll go over to a tree or bush, bite off a small branch and just pulverize it, usually after running about proudly showing off his new toy. What's the downside to this, beside him pruning our landscape, and how can it be stopped?  Is this any indication of a diet deficiency? He has thrown up twice, usually 8-10 hours after ingesting what obviously didn't digest.

Answer
I kind of lean toward let well enough alone.  Scraps of lumber can splinter if he chews them, but the sticks that fall off trees usually crumble if chewed.  The green sticks usually are too flexible to be a danger.  I guess you could try the lawn and garden sprays meant to keep dogs from marking bushes.