Q.brown spots on the leaves of my fruitless mulberry
The leaves get brown spots on them, then the leaves turn yellow and fall off. I live in Texas and we have had record rain fall this spring.

The heavy rainfall may be the cause.
A well established, healthy tree should be able to withstand the flooding rains.
I would watch the tree for any additional signs of disease or insect infestations.
I do think your tree will recover.
Here are a few articles for things to watch for.
https://www.gardeningknowhow.com/ornamental/trees/fruitless-mulberry/possible-causes-of-a-fruitless-mulberry-with-yellow-leaves.htm
https://www.gardeningknowhow.com/ornamental/trees/fruitless-mulberry/fruitless-mulberry-trees.htm