Trees, like any other living thing, can get sick and die. However, it may not always be obvious that they are dead and rotting. Even if you think that the tree is dead, the questions are whether the tree should be removed and if so, when it should be removed.
One of the things that you should consider when deciding whether or not to remove a tree is that it can be part of a natural recycling process. The tree can help to feed wildlife and give them a home. For example, the insects that woodpeckers feed on can breed in sick and dying trees. That can give woodpeckers a good place to feed. An added benefit of this is that if the woodpeckers are going to these trees, they are leaving your healthy trees alone.
You should look at any damage before you decide whether or not it's time to get rid of a tree. If the trunk is noticeably damaged, then you should have some professionals, such as Johnson's Tree Service & Stump Grinding, come out and take it out. The damage that you are looking for includes large chunks of the trunk missing and large vertical cracks. Those things will weaken the tree and make it easier for it to fall over.
Even if the tree isn't noticeably damaged and is providing a good refuge for wildlife, there are some times that you should automatically get the tree removed. Those times include when the tree presents a significant risk of harm or damage. That includes trees that could fall on your house, neighbors' homes, or electrical wires. If the tree could disrupt traffic if it were to fall it should also be removed. Another thing to look at is if the tree is dropping limbs. The falling limbs can present a lesser but still significant risk.
Once the decision has been made to remove the tree, what are you going to do with the dead tree? There are a few things that you can do. One is that you can use it for naturally composting landscaping. If you have the room, you can put parts of the tree into gardens, find plants that will grow on it, and let it decompose into your garden. The tree can slowly enrich the soil and feed your plants. Another option, if you have the space, is to put the felled tree into an out-of-the-way location on your property and let it be used for wildlife, some animals can use it for a home.
Whatever you choose to do with the tree, you need to make sure that you know when you should remove it.