This work introduces a piecewise linear map, defined on three partitions of the unit interval. As a discontinuous generalisation of the flat top tent map, the system function is constant in the middle partition and therefore displays a rich variety of superstable orbits. Symbolic dynamics and interval mapping are used in order to investigate the dynamical properties of this map, including the newly discovered nested period incrementing bifurcation structure. It turns out that the stable periodic orbits are arranged according to an infinite binary tree of the corresponding symbolic sequences, which can be generated by a simple set of rules. The system also allows for straightforward computation of the respective regions of existence in parameter space. The results of this work may also be transferred to other three-partition maps, examples of which are given. This also leads to the conclusion that the famous U-sequence is embedded into this highly structured bifurcation scenario.