Answer: Generally, surface mulches decompose so slowly that they don't create a nitrogen deficiency in the soil. I would imagine that a very thin layer of bark mulch wouldn't cause any harm--but I wouldn't put it more than an inch or two deep. In my mind, that mulch would mimic the leaf litter and other organic matter present in a woodland setting.
That said, I wonder if you rake up and dispose of all the leaves each fall? Those leaves represent a huge amount of nutrients taken up by the trees; again, in a natural setting, these nutrients are "recycled" as the leaves fall to the ground and decompose, and the nutrients are returned to the soil so the tree can use them again. If you've been removing the leaves year after year, you might want to consult with a local arborist about whether you should be fertilizing the trees. Perhaps a layer of compost, spread under the trees each spring, would help replace those nutrients.
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