Putting chippings or plant materials around the base of a tree will not cause it to suffocate from lack of oxygen, or compact the soil. The weight of the materials would not be enough to cause soil compaction, and plant material and chippings are light and porous enough that air can still get to the soil. The root systems of the trees will also probably extend far beyond the boundaries of the mulch. There is, however, a possibility of spreading disease to the trees if the garden materials being put around them have not been thoroughly composted. Fungal spores can be spread from rainfall splashing from ground material to tree leaves, and pests can hide in the dead plants and move out into a new host. If the trees are tall, this is less of a problem, but it's generally not a good idea to put fresh dead plant material around live ones- just in case. The same idea applies regarding composting near trees; many people put compost piles in the back wooded corners of their properties and have no problems. On the other hand, it's an easily avoidable means of spreading disease, and composting should be done in a contained bin away from live plants.