Yep, I'm in Houston.
Yes it fell during a storm, but how could one say it wasn't lightening or natural decay that caused it to fall? The tree snapped at the base. As far as I could tell it didn't have full growth around the base any longer and it lost support enough on one side. One way or another it was going to come crashing down, storm or no storm.
This could be your saving grace.
I would file a claim against the neighbor's homeowners insurance. There's more to the process, but this is the direction I would be going with this loss.
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I'd say decay of the tree played a good role, although it probably wasn't apparent to the owners of the tree prior to the storm. If it had not come down in this storm it would have come down on it's own..
This tree needs to be examined by a professional.
Your argument should be along the lines of "neighbor neglected to remove a decaying tree, thus creating a hazard that ultimately damaged my property. Here is the bill for my damages.."