  We provide a full and unbiased solution to the Dyson-Schwinger equation
illustrated for $\phi^4$ theory in 2D. It is based on an exact treatment of the
functional derivative $\partial \Gamma / \partial G$ of the 4-point vertex
function $\Gamma$ with respect to the 2-point correlation function $G$ within
the framework of the homotopy analysis method (HAM) and the Monte Carlo
sampling of rooted tree diagrams. The resulting series solution in deformations
can be considered as an asymptotic series around $G=0$ in a HAM control
parameter $c_0G$, or even a convergent one up to the phase transition point if
shifts in $G$ can be performed (such as by summing up all ladder diagrams).
These considerations are equally applicable to fermionic quantum field theories
and offer a fresh approach to solving integro-differential equations.
