An alternating permutation of length is a permutation such that . Let denote the set of alternating permutations of , and let be the set of alternating permutations in that avoid a pattern . Recently, Lewis used generating trees to enumerate , and , and he posed some conjectures on the Wilf-equivalence of alternating permutations avoiding certain patterns of length four. Some of these conjectures have been proved by Bóna, Xu and Yan. In this paper, we prove two conjectured relations and .