When a cottonwood tree is cut down, does the stump die, or does it send out shoots that grow into more trees?
And, if a cottonwood tree located on a hillside is cut down, what is the risk of erosion?
As it turns out, some poplars and cottonwoods sucker from the roots and some do not. Determining what kind of cottonwood you have is the key to answering this question.
Identifying tree varieties can be tricky. The best way to get a positive ID is to take a sample to the Hyde Herbarium at the Center for Urban Horticulture (near the University of Washington). It is definitely worth a visit, as it is the only herbarium on the West Coast that serves the public.
Hours, driving directions, how to collect specimens, etc. are on the Hyde Herbarium page.
As for your second question, here is what the Washington State Department of Ecology’s Vegetation Management: A Guide for Puget Sound Bluff Property Owners has to say (p.25):
Given the importance of tree cover on potentially unstable slopes and the advisability of retaining them for erosion control purposes, a landowner should explore alternative options to tree removal or topping…[if a tree must be cut] stumps and root systems should be left undisturbed…[to reduce the risk of erosion].
A companion website from the Washington State Dept. of Ecology contains a great list of groundcovers, shrubs and trees that will help keep your slope intact if you decide to remove the cottonwood. The website includes a Plant Selection
guide.