If you suspect that you have a dead, diseased, or otherwise dangerous tree on your property, most arborists suggest that it be removed before it causes damage to your property or infects any neighboring trees. If it is an imminent and dangerous threat, the tree should be removed immediately.
However, the City requires that you obtain a no-cost tree cutting permit to do so. Email the Planner on Call or call 425-556-2494, stop by the Permit Center on the second floor of City Hall, or print a Tree Removal Application (PDF).