My concern was the chance that birds or squirrels had nested in the tree. For awhile, I debated whether to ask the men if they'd found a nest -- did I really want to know? Well ... of course I did.
They told me they always check before cutting, and if a nest with young is found, they'll wait a few weeks until it's vacant. Unfortunately, sometimes they don't see the babies and nest until it's on the ground. At that point, apparently there's nothing they can do. That's the part I didn't want to know (I'm notorious for sticking my head in the sand). Luckily, there was no nest in this tree.