This work is concerned with a mechanical model of a sympodial tree with first-level branches, which has been shown to exhibit certain properties potentially suitable for biomimetic applications. To investigate these potential benefits further from the viewpoint of the system nonlinear behavior under external periodic excitation, modern numerical tools related to the concept of dynamical integrity are either adjusted or newly developed for this system for the first time. First, multistable regions of interest are isolated from bifurcation diagrams and the effect of damping is investigated. Then, in order to obtain the corresponding basins of attraction of this highly dimensional model, an original computational procedure is developed that includes cell mapping with 406 cells, where each cell represents an initial condition required to construct the map. Full 6D basins are computed, and they are reported for various values of the damping parameter and the excitation frequency. Those basins are then used to calculate the dynamic integrity factors so that the dominant steady-state can be determined. Finally, the integrity profiles are reported to illustrate how the robustness varies by changing the system parameters.