Mushrooms (Corallimorphs) have a few different methods of reproduction it just depends on the family they come from. Here's some info quoted from Julian Sprung & J. Charles Delbeek's "The Reef Aquarium, Vol. 2"
Methods of Reproduction:
*Longitudinal Fission: The polyp splits at the mouth and separates forming two new polyps.
*Pedal Laceration: As the Polyp moves over rocks and substrates, small sections of the pedal base are torn and left behind. These pieces form miniature versions of the mother polyp. (this is the one that sir_dude was talking about)
*Transverse fission: The polyp grows tall and due to current forms a twist point in the column and the top end of the polyp is severed from the rest of the column and pedal base. This top portion floats away and reattaches elsewhere.
All Corallimorphs can do either of these methods, but some of these methods are more common than others in different families. For example:
*Ricordea (Yuma & Florida) - Primarily Longitudinal Fission, Pedal Laceration common, Transverse Fission rarely.
*Discosoma (Blue, Green, Red, Striped, Pimpled, Bullseye) - Pedal laceration is most common, longitudinal fission is possible but uncommon, transverse fission is common under high flow or crowded areas.
*Rhodactis (Hairy Mushrooms) - Longitudinal fission mostly, pedal laceration is possible but uncommon
Hope that helps.