stolon

1a :  a horizontal branch from the base of a plant that produces new plants from buds at its tip or nodes (as in the strawberry) —called also runnerb :  a hypha (as of rhizopus) produced on the surface and connecting a group of conidiophores

2 :  an extension of the body wall (as of a hydrozoan or bryozoan) that develops buds giving rise to new zooids which usually remain united by the stolon