Unfrustration Condition and Degeneracy of Qudits on Trees

We generalize the previous results of [1] by proving unfrustration condition and degeneracy of the ground states of qudits (d-dimensional spins) on a k-child tree with generic local interactions. We find that the dimension of the ground space grows doubly exponentially in the region where rk<=(d^2)/4 for k>1. Further, we extend the results in [1] by proving that there are no zero energy ground states when r>(d^2)/4 for k=1 implying that the effective Hamiltonian is invertible.