The homeowner is liable for the damages and responsible for the tree removal. 
These trees are managed by the City of Bellevue, and if they pose a threat to a homeowners property due to disease, rot, weak root system, etc. it is the responsibility of the home owner to notify the City to have the tree(s) in question removed. Some of these trees are quite tall and their ability to reach out and hit your house are surprisingly high. To report a potentially unhealth tree, call 425-452-6855 or email parksweb@bellevuewa.gov

Tree "ownership" is determined by whose property the trunk stands completely on. If the tree is a boundary tree (the trunk straddles a property line), it is owned jointly. If most cases, property owners may trim branches and roots which encroach on their property if the stary with certain guidelines: trim only up to the property line; do not enter the owner's property without permission; do not destroy or damage the tree by trimming roots or branches.

If you have a dispute with your neighbor, call the Mediation Program at 425-452-4019. they can help you explore your alternatives, coach you on how to negotiate with your neighbor or provide free mediation to help you and your neighbor find a workable solution.

One of the leading reasons given for tree removals is safety. To ensure you are needlessly removing an otherwise healthy tree, it pays to talk with a qualified certified forester. Ther are two websites that can help. Contact the International Society of Arboriculture at www.treesaregood.org or the Society of American Foresters at www.safnet.org.

For most homeowners, the answer is no. The 41.5 Covenants have specific language about this that should be reviewed and by contacting the VP/Architecture board official. The City of Bellevue official document on tree removal is here
From a Bellevue City perspective, a permit may be required if one or more of the following conditions is true:

• Tree removal results in more than 1000 square feet of disturbance of the ground.
• the property is part of a designamted critical area or buffer.
• The tree is in a native growth protection area or a retained vegetation area.
• The tree is on the city's right of way adjacent to your property

Contact the Land Use desk in Development Services (425-452-4188) or landusereview@bellevuewa.gov for assistance in determining whether you need a permit.

Contact Permit Processing in Development Services at 425-452-4898 or permitprocessing@bellevuewa.gov

Disturbance of the ground results from any action that displaces or destroys vegetation, resulting in increased soil exposure. This includes exposure of the existing ground due to tree removal, where no other vegetation is growing under the tree. Disturbance can also result from the physical process of tree removal, where foot and vehicle traffic, construction of access roads, felling of trees, dragging of felled trees and excavation of stumps disturb the ground.