Before revision:  We show that in this setting, the process has two different time scales: the time to the first event grows to infinity, but the inter-event times nearer the root of the tree are approximately exponentially distributed.
After revision:  We show that in this setting, the process has two different time scales: the time to the first event grows to infinity, but the inter-event times nearer the root of the event times are the order statistics of n logistic random variables with mode \log(1/\psi); moreover, we show that the inter-event times are approximately exponentially distributed.