First of all, we are talking of incremental rotary encoders, rotating with speeds that requires high-speed processing by PLC.
A single direction counting requires only one pulse train from encoder to PLC.
The quadrature mode means using two phase shifted outputs of the same encoder and allows a PLC to recognize the encoder rotation direction, consequently changing the counting direction as necessary.
Not only applications with reversive movement do require this mode.
It may be necessary, for instance, in indexing application without reset, if stopping generates back-and-forth vibrations or just a single rollback. The extra pulses will be counted in wrong direction if a single direction counter is used.
The hardware reset is another pulse from encoder to PLC.
It appears in only one specific position of the encoder and resets the encoder counter value in PLC.
It is used as a zero position reference.
If such reference is taken by software from another part of equipment, you do not need a hardware reset for encoder counter.
Each pulse from encoder requires one high-speed PLC input.
Therefore, a single phase high-speed counter occupates one high-speed input, while a quadrature counter with hardware reset occupies three.
I am not familiar with Mitsubishi A1SH, but Mitsubishi FX1N and FX2N series PLCs have 6 high speed inputs.
They may be assigned to one high-speed count or reset each, with maximum of 2 quadrature counters.