River birch trees drop leaves as soon as the weather heats up and dries out. Regardless of how much you are watering, unless it is in a wet location, it will drop leaves. I think the difference in your two trees, is one seems to be nearer the building and concrete which limits the root system, while the other has more yard to grow in, and take up moisture in. This is one of the reasons, I often don't recommend planting them in a typical yard. It shouldn't hurt the tree, provided you are watering some.
Redbuds do great in full sun or partial shade.
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