When a tree falls on both your property and neighbors who is responsible by law?

It depends on the laws of your country.
In the United States at least, no one is held liable for acts of nature, so you would each deal with your bit of the damage to your property (claim on your home owners insurance).

In the UK, the owner of the tree would likely be liable. That would be the owner of the ground on which the tree was growing. Again the thing to do is claim on your home owners insurance and leave it to the insurance company to decide who was responsible.

Also, See related questions below.