This will depend on a few factors.
Every City or County has it's own different set of laws and ordinances for tree removal. You will need to refer to your local cities tree preservation or landscape ordinance for rules and regulations.
Most tree preservation ordinances will stipulated any tree with a trunk diameter smaller than X" can be removed without a permit. Again, you will need to refer to your local cities tree removal permit regulations.
Every city has a list of trees that it considers a local pest and may be removed without a permit no matter the size.
As a general rule, you do not need a permit to remove a dead tree or a tree that is posing an immediate danger to persons or property. Ask a qualified arborist to come out and inspect the tree. They will soon tell you if you can remove it.