  We present the construction and stochastic summation of rooted-tree diagrams,
based on the expansion of a root finding algorithm applied to the
Dyson-Schwinger equations (DSEs). The mathematical formulation shows superior
convergence properties compared to the bold diagrammatic Monte Carlo approach
and the developed algorithm allows one to tackle generic high-dimensional
integral equations, to avoid the curse of dealing explicitly with
high-dimensional objects and to access non-perturbative regimes. The sign
problem remains the limiting factor, but it is not found to be worse than in
other approaches. We illustrate the method for $\phi^4$ theory but note that it
applies in principle to any model.
