It doesn't matter if the tree is a nuisance, damaged, or a hazard -- tree removal can be a dangerous process. In most cases, this isn't a job for a homeowner but one that is best left to a professional due to the inherent dangers of taking down a tree.
It may seem like simple work to cut down a young tree that is barely taller than yourself, but the process becomes a lot more difficult and dangerous if the tree is tall. It's harder to plan for the direction in which a taller tree will fall, particularly if one isn't trained in tree felling. There is also more risk for injury, as one typically has to climb into a taller tree to limb or section it before removal.
It can be easy to predict the direction of fall for a leaning tree, on the other hand, as gravity will tend to take over and bring the tree down on the side of the lean. Unfortunately, this can also be a problem if there is a structure or other obstruction in the path of the tree's likely fall. Professional tree technicians will use cabling and braces, along with sectional removal techniques, to remove leaning trees without damage.
Location matters. The closer a tree is to other structures, roads, or power lines, the more dangerous and challenging it is to remove. Trees that grow into or near power lines, for example, may require that the power is temporarily turned off during removal. A technician may also need to get into the tree to carefully cut off branches and lower them to the ground so they don't pull down power lines or fall on your house.
Density affects tree removal in two ways. Trees with a dense crown are heavier and more unwieldy to take down, for example, so heavy pruning of the crown is needed before safe removal can proceed. The other density issue is when several trees grow closely together. It can be nearly impossible for an amateur to remove one tree from a dense copse without damaging the nearby trees, so it's best to call in a service.
Living trees can be safer to bring down than dead or badly diseased trees, simply because there are fewer brittle branches that can break off during the process. For safety, a service tech may use special equipment, such as lift trucks and climbing safety harnesses, to remove branches from unhealthy or dead trees. You likely won't have the benefit of this equipment if you attempt removal yourself.
Contact a tree removal service for a safer alternative to attempting to do it yourself.