There was a hypothesis proposed recently that there are living things from another tree of life kicking around but we don't find them because we're not looking for them in the right way. I tend to think not, but, hey, the thought is out there.
The problem for any new starting life is that it has to contend with existing life that can, with the greatest of ease, just eat it. The first replicators were fragile, ineffectual things ill-prepared to contend with the streamlined, highly evolved living machines of today - or even 100,000 years after the first replicator.
Eukarya involves a merging of Archaea and Bacteria. As is entirely obvious from the genetic of DNA transcription and replication, as well as the endosymbiotic origins of mitochondria. Whether they evolved from Archaea, or from a merging of an archaeon and a bacterium (the fusion hypothesis) is not yet determined, although I favour the latter.
Archaea and Bacteria too share common ancestry as is, again, obvious from their systems of DNA transition and replication (in particular there is a 14 nucleotide sequence in rRNA conserved across all life on earth). And, here, in fact things are more complicated because there is a certain amount of horizontal gene transfer between the two domains.
Iunno. I tend to see the three domains as separate trees, considering how much they have diverged even if they shared common ancestry.
So... even though they all share a common ancestor, you're going to view them as different trees of life anyway? o_O
I can only think you're using a different meaning of "Tree of Life" to the rest of us. In any case it's not a particularly useful view (any more than it's useful to view different phyla as different "trees of life"), firstly because the homologies between the different domains mean that discoveries from one domain can be transferred to others (for example, discoveries of certain proteins in Archaea have been used to search for homologous proteins in Eukarya and then, using the archaeon as a model, to understand what the proteins do in the eukaryote) and, secondly, because the borders between the domains are not clear cut: horizontal gene transfer occurs between Archaea and Bacteria, and organelles such as mitochondria and chloroplasts have their endosymbiotic origin in Bacteria, meaning Eukarya cannot be fully understood without investigating other domains.
Somebody mentioned viruses. Viruses have their own trees of life, and for that reason some biologists exclude them from being "life." But I would count them as life.
Most biologists do not count viruses as life, but not for that reason. Rather they're not normally considered life because they have no metabolism and are incapable of reproducing themselves but instead only capable of inducing others to reproduce them for them.