Concept says, by default all switchports are in trunk mode. So if any switch is connectd to a port, it tries to negotiate the trunk & once established, send BPDUs. Thus, all access ports have portfast turned on which denies any BPDUs received on port.
Coming to your point, Yes uplinks will share BPDUs. If ur topology has redundant connections, then you are prone to loops if stp is turned off. However, if ur only concern is to limit the diameter of stp, prefer using "vlan allowed" comand on trunks for stp to limit to specific vlans & thus not flooding entire network.
