This is not reingold-tilford. The algorithm has two parts. The first part is a heuristic to distribute nodes into multiple layers when it won't fit in one. Then for each layer, it is a physics simulation with the labels. Each label has attraction force to the desired position, but they also push each other.
Reingold–Tilford was referenced below but doesn't seem applied in this case.
I've attempted to look through the code here but comments are limited and it is pretty imperative.