Hello CA Mom and thanks for your post.
Depending on whether the tree is considered a "heritage" or "protected" tree, the process can be easy or difficult. A "heritage tree" or "protected tree" is defined, not by its height, but by the width of the trunk at a specific point above ground. In the case of Sunnyvale, measure the trunk of the tree at 4.5 feet from the ground and if the width of the trunk is greater than 38 inches, a permit (costing $223 each) and City permission will be required to remove the tree. If the tree trunk is smaller than 38 inches, you may pull the tree without permission despite the height.
If the tree is located in the backyard of the home and is NOT deciduous (meaning the leaves fall off in the winter and bloom again in the spring), then the chances for City approval are pretty good. Size of the tree, light, root invasion and "falling" or "dropping" hazards are all taking into consideration when evaluating a tree. In the case of the Magnolia (especially the "Saucer Magnolias" and "Liquid Ambers" that the City planted in all the frontage strips in Sunnyvale eons ago), we now know that these have extremely invasive roots and are being pulled in favor of other tree species. Similarly Semper Virens (redwoods) are often poorly placed--too near to irrigation systems, and, as a result, they grow too fast and too tall for the area. All of this is taken into review when the City arborist checks the application. To help your application process, plan to obtain the professional opinion and recommendation of a certified arborist to submit with your application.
One more thing, most cities now require at least "one" tree to be planted on a city sized lot of 6000-6500 square feet. If these are the only two trees at your home (not including any street frontage tree), then the arborist may require replanting at least one deciduous tree to ensure adequate air exchange at your home. So be prepared that, if you want permission for both trees to be removed, you may need to offer to install one deciduous tree in its place. There are many varieties of deciduous trees that are both beautiful and small enough to prevent light intrusion in the future--again, talk with your arborist for assistance.
Grace Morioka, SRES
Area Pro Realty
Santa Clara County
Tel (408) 426-1616