Something else that nobody mentioned. Usually if the tree is in a wet area, the root system may not be so deep. If the tree is in a dry area, the root system is probably deeper. A shallow root system on a dead tree would probably rot pretty quickly while a deeper root system would be good for a few more years.
It depends on the specific variety of tree.
Different trees decay at different rates: some are strong and sturdy. Good lumber often can be salvaged
10 years after a forest fire kills a stand of trees for good types.
On the other hand, some species rarely attain maturity without the core already starting to decay.
Then there is the root structure: that is determined more by the type of tree than the amount of water
in the soil. For example, the Coast Redwoods of California have a deep tap root - they can live in solid
groves and grow to 350+ feet before the wind blows them over. By contrast, the Sierra Redwoods have
only a surface root system - they have to ground in a mixed grove with pines and firs to provide a
wind break, as they will blow over if they get much above the surrounding treetops.
Personally I'd feel much more comfortable mounting a beam on a Sitka Spruce than on a Cottonwood,
but those are trees that I'm familiar with. As you can see from the redwood example, even closely
related species can have significant differences in growth characteristics, which is why you need to
know the actual type of the tree.