A data processing apparatus contains branch prediction circuitry including a micro branch target buffer, a full branch target buffer and a global history buffer. The branch target buffer entries contain history data which indicates whether or not a number of the following blocks of program instructions, subsequent to and sequential to a block of program instruction identified by that branch target buffer entry containing a branch instruction, do themselves contain any branch instructions. If the history data indicates that the following blocks of program instructions do not contain branches, then the operation of the branch prediction circuitry is suppressed for these following blocks of program instructions so as to save energy.