As to the lawn model: Is it likely that there was first a living organism and that everything living descended from it?
My understanding is that fossils generally cannot provide tissue to map DNA information. If that is the case, and if this research project stays away from making projections based on evolution theory, what they'll end up with is a huge database with specific genetic information (which is also huge) on each and every known type of creature - plus a few that are now extinct.
Therefore, I do not see where this can project will have the database to graph a "tree" which has time vertically and morphology horizontally. It would be based on available genetic information.
But using that information, I can see them sorting, parsing and matching the database to look for genetic matches. If the resulting chart is genetic information vertically and morphology horizontally - I personally think it'll look like a "lawn."
Just my two cents...
Therefore, I do not see where this project will have the database to graph a "tree" which has time vertically and morphology horizontally - because the graph would be based on available genetic information as opposed to fossils.