The branches of my Santa Rosa plum tree have these little black/brown
|What you are describing are scale insects. They can cause problems if the population gets quite large. The first step in scale control is to apply a dormant oil spray in the spring before growth starts. The dormant oil spray should be followed by an all-purpose fruit tree spray or the insecticide Imidan. Follow label directions - additional sprays may be needed later in the growing season. Since you've missed the window of opportunity for a dormant oil spray this year, you can wait until next winter or early spring and put your plum tree on a regular spray schedule to control the pests.
Best wishes with your tree!