It is not completely understood how a liver can grow back to its original size and function.

The liver has a remarkable capacity to regenerate after injury and to adjust its size to match its host. Within a week after partial hepatectomy (surgical removal of all or part of the liver), which means normally the surgical removal of two-thirds of the liver, hepatic (liver) mass is back essentially to what it was prior to surgery. This observation has prompted considerable research into the mechanisms responsible for hepatic regeneration.

The Dynamics of Liver Regeneration
Researchers found that partial hepatectomy leads to proliferation (or growth) of all populations of cells within the liver, including hepatocytes, biliary epithelial cells and endothelial cells. DNA synthesis is initiated (or started) in these cells within 10 to 12 hours after surgery and essentially ceases in about 3 days. It appears that hepatocytes (fragments of the liver) have a practically unlimited capacity for proliferation (growth).

What stimulates the Regeneration?
Hepatic regeneration is triggered (or started) by the appearance of circulating mitogenic factors. >As might be expected, liver regeneration seems to be supported by a group of mitogens and growth factors acting in concert on several cell types. Some of the major and well-studied players that act together in this process include:
Hepatocyte growth factor
Interleukin-6
Epidermal growth factor
Insulin
The processes and signals involved in shutting down the regenerative response are less well studied than those that stimulate it.