Trees may pose a hazard even though they're alive. Consider the potential threat in terms of the tree health, any target that it might damage if it fell and the value of the target. Trees near homes and utility lines are a particular concern.
Some defects in trees are obvious, but a qualified expert, such as a certified arborist, can take a core sample to check for deterioration. He or she may have suggestions, such as pruning, to mitigate the hazard without losing the tree.
Check municipal regulations about tree removal and disposal. A skilled technician, such as an arborist, has the equipment and skills to cut the tree in a controlled, safe manner. The technician will usually dispose of it by mulching, which can be kept for mulch. The arborist can use a stump grinder to remove the rest of the tree, or it can be dug out. The hole that remains should be filled in.