Until and unless state law is changed, in Vancouver you are responsible for all the maintenance of street trees, even though they are on city property (the first ten feet from curb toward house). You are also responsible for sidewalk maintenance. Please do this maintenance. If someone trips on your sidewalk because you haven't maintained it, you may be held financially responsible for the injuries. Likewise if a branch is broken off and left in the traffic you might be held responsible, so it pays to take advantage of the annual cleanup for yard debris, including branches and special opportunities such as that provided after the January 2004 storm for free disposal of tree branches or the tentatively annual wood chipper event. Also, please note that if you have a valuable tree, such as a cherry or walnut that has sufficient value but needs to be removed, you should consider selling the log to a small, local mill or to a woodworker. However, if the tree is a street tree (witnin 10 feet of curb), unless the city provides a waiver of some kind after a storm, a $5 permit is required to remove it and the city provides advice on what may be planted that is least likely to disrupt your water line, your sewer line and the sidewalks and the street. Contact the city forester. Removed street trees (those within ten feet of the curb) need to be replaced unless the city waives that requirement in your special situation.
Also note that even severely damaged trees can often be saved. The author of this page cut all of the branches off a variety of cherry tree planted too close to the house, yet the tree grew back a full set of branches. You can see what happens or consult an arborist as long as the main body of the tree is structurally sound. (You don't want to be responsible for half a tree falling on someone's car or house.)