  We investigate the effective resistance $R_n$ and conductance $C_n$ between
the root and leaves of a binary tree of height $n$. In this electrical network,
the resistance of each edge $e$ at distance $d$ from the root is defined by
$r_e=2^dX_e$ where the $X_e$ are i.i.d. positive random variables bounded away
from zero and infinity. It is shown that
$\mathbf{E}R_n=n\mathbf{E}X_e-(\operatorname
{\mathbf{Var}}(X_e)/\mathbf{E}X_e)\ln n+O(1)$ and $\operatorname
{\mathbf{Var}}(R_n)=O(1)$. Moreover, we establish sub-Gaussian tail bounds for
$R_n$. We also discuss some possible extensions to supercritical Galton--Watson
trees.
