Targum Onkelos on Breishit 2:9 translates the term עץ החיים as meaning "the living tree in the middle of the garden." Based on this translation, there is no name given to this tree. And, you will notice that Adam was not warned not to eat from this tree either - at least not at this point.
Kil Yakar commentary explains why. In the Garden of Eden, all trees had a taste. It was also part of creation that the fruit of the tree and the tree itself had the same taste. Kli Yakar explains that had Adam eaten from this specific tree, he would have lived eternally. However, this tree, had no beautiful fruit. And, G-d was not concerned that humans (or any animals for that matter) would eat from a plain tree having no fruit to attract him to it.)
We can speculate what type of tree this may have been. Considering that there are few trees, currently, where the bark is edible (only two I can think of are sugar and bamboo) this may have been a species that no longer exists. The Torah doesn't indicate its species, and none of the commentaries I have searched indicate this, either.