AbstractAcyclic networks are represented by rooted treesTfrom a given simply generated family ℱ. Let ℱn,pbe the subset of ℱ containing all treesTwithnnodes andpleaves. Assume thatTis selected uniformly from ℱn,pand that each edge ofthas probabilityqof failing. LetZi= 1 if the path conecting the root ofTto the ith leaf does not contain a failed edge, O otherwise. We investigate the stochastic process (Zi,…,Zp). Asympototic results, manly for the family oftary trees, are derived. As auxiliary results, a general rotation lemma for simple families of trees is given, and the (joint) asymptotic leaf height destributions in tary trees determined. © 1994 John Wiley