Strictly speaking, there is still some cell division shortly after birth and into the early phases of post-natal development. So the brain increases in size, slightly, via that method. But mostly it is from the elaboration of the dendritic arbors of the existing neurons. The size and branching factors increase many times during development. Check out the beautiful (and stunningly accurate) drawings of the famous neuro-anatomist Santiago Ramon y Cajal.

For example, the Purkinje neurons of the cerebellum:

http://www.gladstone.ucsf.edu/wp/2009/06/knowledgegrows/

Note how the number of branches increases to sample more of the volume.