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Hello and thank you for using this site to help you. There are plants that are toxic to dogs, in some cases it is the root, the leaves, the stem/trunk or all parts. Bushes such as Azalea would not be good to eat.
So it is important for you to know what you have in your yard and go over the toxic list which I will put a link to below.
The other problem with tree barks and branches is that they can be swallowed in large or sharp pieces. A large piece can cause an obstruction of the intestinal tract, a sharp piece can actually pierce the intestinal tract.
So if the plant or limb is not toxic to the puppy then you need to monitor what he is taking in. How big or sharp are the pieces, is he swallowing them or spitting them out, etc.
Here is a list of toxic plants and bushes