Memristors are passive non-linear circuit components with memory characteristics, and have been recognized as the fourth basic circuit component, along with resistors, capacitors, and inductors. It has been nearly half a century since the conceptualisation of the memristor, and related research has mainly focussed on the two aspects of binary and continuous memristors. However, compared with these two types of memristors, tri-state and multi-state memristors have greater data density per device, with rich dynamics and great potential in logic and chaotic circuit applications. Moreover, previous studies show that the series-parallel connection of memristor generates more diverse circuit behaviours and increased capacity over a single memristor. However, most of this research is based on mathematical analysis, and lack behavioural circuit simulations or experimental validation. Here, the tri-state memristor is proposed and the mathematic and equivalent Spice models of the tri-state memristor is shown. Furthermore, the circuit characteristics are studied with a complete characterisation of its series-parallel behaviours of the tri-state memristor. Simulations are performed with LTSpice, and the results verify the theoretical analysis, which provides a strong experimental basis for the study of combinational memristive circuits.