is it possible to completely reconstruct a family tree by DNA comparison alone?
No, it is not possible to determine ancestry with just two DNA samples, but it is possible with three: http://www.nature.com/nature/journal/v487/n7407/full/nature11251.html.

Essentially, each sequence region in the progeny can be traced to either the mother or the father, but you really need both sequences to have any confidence.  That would get you one node on your tree.  You would need additional samples from higher generations to rebuild an entire ancestry.

However, it might be possible with hundreds of samples from laterally related individuals.  In fact, I think this is the fundamental hypothesis of the 1000 genome project, http://en.wikipedia.org/wiki/1000_Genomes_Project, and it pretty much worked.  Isolated populations can be discriminated by broad DNA sequence analysis.  

To reconstruct your tree for a single ancestry, you would just want to use more closely related individuals (many, many of them), and then generate a phylogenetic tree, which should roughly correspond to the family tree you are investigating.