Strong dynamic loads in man-made slender structures are likely to produce large amplitude vibrations of due to their flexibility, which often cause functionality losses or inflict irreversible damages. The amplitude of motion results, on one hand, from the characteristics of the loading, and on the other hand, from the characteristics of the structure in terms of inertia, stiffness and damping. A high rate of damping, i.e. mechanical energy dissipation whatever the physical mechanism, is a standard way to limit vibrations of large amplitude and their undesirable consequences. However, adding damping in slender structures remains a challenging problem in engineering. As slender and flexible structures, trees repeatedly endure strong climatic events, yet with minor or no damages in most cases. Therefore, trees regularly submitted to natural flow excitations by wind are likely to possess efficient and specific strategies of damping. In fact, several damping mechanism have been identified in trees: viscoelastic behaviour of wood, aeroelasticity from the interactions with air, and the linear tuned-mass-damper effect between the trunk and the branches. However, the latter, as a linear mechanism, is not amplitude-dependent and therefore does not damp specifically large amplitude motions. A new amplitude-dependent damping mechanism, inspired from trees, is identified and characterised here in the simplest tree-like structure, a Y-shape branched structure. Through analytical and numerical analyses of a simple two-degree-of-freedom model, branching is shown to be the key ingredient in this protective mechanism. This damping-by-branching originates in the geometrical nonlinearities and is therefore specifically efficient to damp out large amplitudes of motion. A more realistic model of the Y-shape structure, using beam finite-element approximation, exhibits the same effect, which seems to be robust. Similarly, two arbitrary multi-branched slender architectures show significant levels of damping of the trunk motion while only the relative branch motion is damped. Finally, the same numerical analysis is performed on a digitalised walnut tree geometry showing the significant damping contribution of this mechanism during large amplitude motions.