Many applications in parallel processing have to traverse large, implicitly defined trees with irregular shape. The receiver initiated load balancing algorithm random polling has long been known to be very efficient for these problems in practice. For any ε > 0, we prove that its parallel execution time is at most with high probability, where T _rout, T _split and T _atomic bound the time for sending a message, splitting a subproblem and finishing a small unsplittable subproblem respectively. The maximum splitting depth h is related to the depth of the computation tree. Previous work did not prove efficiency close to one and used less accurate models. In particular, our machine model allows asynchronous communication with nonconstant message delays and does not assume that communication takes place in rounds. This model is compatible with the LogP model.