A new simplified formula is presented to characterize genuine tripartite entanglement of (2 2 n)-dimensional quantum pure states. The formula tus out equivalent to that given in (Quant. Inf. Comp. 7(7) 584 (2007)), hence it also shows that the genuine tripartite entanglement can be described only on the basis of the local (2 2)-dimensional reduced density matrix. In particular, the two exactly solvable models of spin system studied by Yang (Phys. Rev. A 71 030302(R) (2005)) are reconsidered by employing the formula. The results show that a discontinuity in the first derivative of the formula or in the formula itself of the ground state just corresponds to the existence of quantum phase transition, which is obviously different from the concurrence.