Eating wood is a very bad habit and should NOT be allowed to continue. The litany of problems that could arise from this behavior are well documented in countless cases (ask any vet). From the small (splinters in the mouth and esophagus) to the large (perforated intestines, stomachs, or obstructed bowels), the problems with this behavior are simply too numerous to list. You may have had a dog that ate wood without problem, but there are many counter-examples. Consider yourself extremely fortunate. It's like saying, "Well, I had a cousin that swallowed batteries his whole life and he never had any problems so it must be alright." Your cousin may have been fortunate enough to avoid the disasterous consequences of this behavior but as a general rule, swallowing batteries should be frowned upon.
You need to stop your dog from doing this. It is basically a teething/boredom habit. The first thing to do is to remove the wood items from his environment. Pick up your branches in the yard, don't give him access to the wood pile, etc. Additionally, give him things that it is OK to chew on - chew toys, etc. - things that are specifically designed to be chewed on by dogs. The next thing to do is to keep him occupied with other things. It's a bored dog that spends his days chewing on wood. If that is not enough to curb the habit, you need to add negative reinforcement from the enviroment. Putting Tabasco sauce or bitter apple on the bottom of the trees, putting an electric fence around the wood pile, or using a shaker can when he goes to chew on the undesirable items. Lastly, you could include negative reinforcement from you - saying "NO" when he decides to begin chewing on a wood item. The most important thing to do however is to take the offending wood out of his reach. If there are trees all over the yard, then you simply can't let him out in the yard unsupervised. It may seem like a pain to you but it beats the alternative - a $1,000 dollar vet bill.
The only things that should be eating wood are termites. Don't let your dog do it. You may live to regret it (and he might not).
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