The partnership firm can continue to operate if the remaining partner/s decide to continue the operations. With this, the old deed with get terminated and will be replaced by the new deed based on the new terms and conditions as decided upon by the new partners.
In response to the first part of your question, there must be atleast two persons to form a partnership. If one of the partners die and there is no legal heir or the legal heir don't want to join the firm, then the partnership firm can no more continue and it will turn into sole proprietorship. This might not be scenario, if the original partnership firm has three partners, wherein, even if one of the partners dies, then also the firm can continue as partnership firm due to fulfilment of basic condition as per the definition of partnership.