The authors consider a class of AND-OR tree circuits and study their response to random-pattern inputs as the depth of the tree is allowed to increase indefinitely. Each binary input of a circuit is independently chosen to be one (zero) with probability x (1-x). The logic of the circuit determines the probability of success (one) at the output as a monotonically increasing S-shaped function of x called the probability transfer function. The probability transfer function of an AND-OR tree is shown to have just one interior fixed point (with respect to changes in depth of the tree) in the