a branch of subterranean origin that finally rises out of the ground. The Raspberry multiplies in this way.

Offset,

a short, prostrate-rooting branch with a tuft of leaves at the end (Houseleek).

Runner,

a long, prostrate-rooting branch with tuft of leaves (Strawberry).

Stolon,

a branch that curves downward and takes root. The Currant multiplies in this way.