  Hierarchical structures are ubiquitous in human and animal societies, but a
fundamental understanding of their raison d'\^etre has been lacking. Here, we
present a general theory in which hierarchies are obtained as the optimal
design that strikes a balance between the benefits of group productivity and
the costs of communication for coordination. By maximising a generic
representation of the output of a hierarchical organization with respect to its
design, the optimal configuration of group sizes at different levels can be
determined. With very few ingredients, a wide variety of hierarchically ordered
complex organisational structures can be derived. Furthermore, our results
rationalise the ubiquitous occurrence of triadic hierarchies, i.e., of the
universal preferred scaling ratio between $3$ and $4$ found in many human and
animal hierarchies, which should occur according to our theory when production
is rather evenly contributed by all levels. We also provide a systematic
approach for optimising team organisation, helping to address the question of
the optimal `span of control'. The significantly larger number $\sim 3-20$ of
subordinates a supervisor typically manages is rationalised to occur in
organisations where the production is essentially done at the bottom level and
in which the higher levels are only present to optimise coordination and
control.
