I am no expert, but I think it all boils down to the fact that life alters its environment. Eating other organisms is only one way of doing so; for instance, the atmosphere where the first organisms developed was quite different from how it is today. There was much less oxygen in the air when the first organisms developed, and when some managed to produce oxygen as a byproduct they caused a massive extinction for those that could not cope. This eventually paved the way for the much more active style of an aerobic metabolism, but another "Tree of Life" might need to develop in an anaerobic environment.
That is, I suspect, why we won't find something like a mouse from a different tree. The transition to an oxygen-rich atmosphere was fast enough to cause mass extinctions but with some organisms surviving and adapting the first time, but a tree that developed in an isolated environment only to be exposed after the transition would just be wiped out.