Yes, but that arrangement will have already been made at the beginning of the trust. That is why either a minimum of two current trustees must exist, or one current trustee and one successor trustee. If one current trustee passes away, then the other current trustee takes over, or the successor trustee takes over.

Then a new co-trustee to replace the lost trustee must be appointed. This person would be appointed by the existing trustee after getting the consent of whoever had played the role of the grantor. Or, if the client who had the trust set up in the first place IS the trustee, then that trustee needs to consult no one about the decision of who will be the replacement trustee. It is his or her own decision.