  In this article we use the Bellman function technique to characterize the
measures for which the weighted Hardy's inequality holds on dyadic trees. We
enunciate the (dual) Hardy's inequality over the dyadic tree and we use the
associated "Burkholder-type" function to find the main inequality required in
the proof of the result. We enunciate a "Bellman-type" function satisfying the
required properties and we use the Bellman function technique to prove the main
result of this article. We prove the optimality of the associated constant with
an explicit example of an extremal family of maps. We also give an explicit
interpretation of the corresponding Bellman function in terms of the theory of
stochastic optimal control.
