Revocable Living Trust Agreement My mother just past away.

Revocable Living Trust Agreement
My mother just past away. My father prepared a Revocable Living Trust Agreement. He lives in the State of California. Once the Trust has been signed and notorized does it need to be filed with the State of California? If so, where will it be filed? Does each of his assets (Bonds, CDs, checking, savings, grant deed on the house and mobile home, etc) need a copy of the Trust so that asset can be added to the Trust?