We extend the well-known Tree-Diffie-Hellman technique used for the design of group key exchange (GKE) protocols with robustness, i.e. with resistance to faults resulting from possible system crashes, network failures, and misbehavior of the protocol participants. We propose a fully robust GKE protocol using the novel tree replication technique: our basic protocol version ensures security against outsider adversaries whereas its extension addresses optional insider security. Both protocols are proven secure assuming stronger adversaries gaining access to the internal states of participants. Our security model for robust GKE protocols can be seen as a step towards unification of some earlier security models in this area.