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

Unfrustration Condition and Degeneracy of Qudits on Trees