environment - Does a plant have to be alive for its roots to prevent erosion?

One argument often given in favour of saving current and planting additional plants/saplings is the roots hold soil together and prevent soil erosion.

Unlike plants that may be uprooted, a large tree may only be chopped down; its roots continue to remain buried. Do these, now dead, roots continue to prevent soil erosion, or does the plant have to be alive and kicking?

My guess is the latter, perhaps because absorption of water possibly 'compresses' the soil over the root.