  We present various results concerning the two-weight Hardy's inequality on
infinite trees. Our main scope is to survey known characterizations (and
proofs) for trace measures, as well as to provide some new ones. Also for some
of the known characterizations we provide here new proofs. In particular, we
obtain a new characterization based on a new reverse H\"older inequality for
trace measures, and one based on the well known Muckenhoupt-Wheeden-Wolff
inequality, of which we here give a new probabilistic proof. We provide a new
direct proof for the so called isocapacitary characterization and a new simple
proof, based on a monotonicity argument, for the so called mass-energy
characterization. Furthermore, we introduce a conformally invariant version of
the two-weight Hardy's inequality, we characterize the compactness of the Hardy
operator, we provide a list of open problems and suggest some possible lines of
future research.
