Inside Beaverton city limits
The tree between the sidewalk and the street is a street tree. You maintain it, and removing it takes a city permit and normally a replacement from the city's approved list. Cutting 20 percent or more of the crown or roots counts as removal. Street trees have to clear 8 feet over the sidewalk, 12 feet over the street and 14 feet over arterials, and topping them is prohibited.
Most ordinary backyard trees need no permit. The exceptions are the trees the development code protects: significant and historic trees, trees in significant groves and natural resource areas, and landscape or mitigation trees that were kept or planted as a condition of the subdivision. If your lot backs onto a greenway, a creek or a preserved grove, treat the trees along that edge as protected until the Planning Division (503-526-2420) says otherwise.