In this article, we study concave recursions on trees, which appear widely in information theory through algorithms such as belief propagation, and in statistical mechanics through models on tree-like graphs, including the Ising model, percolation, and more generally, the random cluster model. These tree recursions can, in fact, be compared with non-linear conductances, or $p$-conductances, between the root and the leaves of the tree. In this article, we estimate the $p$-conductances of $T_n$, a supercritical Galton--Watson tree of depth $n$, for any $p>1$, for a quenched realization of $T_n$. In particular, we find the sharp asymptotic behavior when $n$ goes to infinity, which depends on whether the offspring distribution admits a finite moment of order $q$, where $q=\frac{p}{p-1}$ is the conjugate exponent of $p$. We then apply our results to the random cluster model on $T_n$ (with cluster weight parameter in $(0,2]$ and wired boundary condition) providing sharp estimates on the probability that the root is connected to the leaves. As an example, for the Ising model on $T_n$ with plus boundary conditions on the leaves, we find that, at criticality, the quenched magnetization of the root decays like: (i) $n^{-1/2}$ times an explicit tree-dependent constant if the offspring distribution admits a finite third moment; (ii) $n^{-1/(α-1)}$ if the offspring distribution has a heavy tail with exponent $α\in (1,3)$.