  As a model of trapping by biased motion in random structure, we study the
time taken for a biased random walk to return to the root of a subcritical
Galton-Watson tree. We do so for trees in which these biases are randomly
chosen, independently for distinct edges, according to a law that satisfies a
logarithmic non-lattice condition. The mean return time of the walk is in
essence given by the total conductance of the tree. We determine the asymptotic
decay of this total conductance, finding it to have a pure power-law decay. In
the case of the conductance associated to a single vertex at maximal depth in
the tree, this asymptotic decay may be analysed by the classical defective
renewal theorem, due to the non-lattice edge-bias assumption.
  However, the derivation of the decay for total conductance requires computing
an additional constant multiple outside the power-law that allows for the
contribution of all vertices close to the base of the tree. This computation
entails a detailed study of a convenient decomposition of the tree, under
conditioning on the tree having high total conductance. As such, our principal
conclusion may be viewed as a development of renewal theory in the context of
random environments.
  For randomly biased random walk on a supercritical Galton-Watson tree with
positive extinction probability, our main results may be regarded as a
description of the slowdown mechanism caused by the presence of subcritical
trees adjacent to the backbone that may act as traps that detain the walker.
Indeed, this conclusion is exploited in \cite{GerardAlan} to obtain a stable
limiting law for walker displacement in such a tree.
