\indent The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length $n$ that can be sorted by passing it twice through a stack (where the letters on the stack have to be in increasing order) was conjectured by West, and later proved by Zeilberger. Goulden and West found a bijection from such permutations to certain planar maps, and later Cori, Jacquard and Schaeffer presented a bijection from these planar maps to certain labeled plane trees, called beta(1,0)-trees. \indent We show that these labeled plane trees are in one-to-one correspondence with permutations that avoid the generalized patterns 3-1-4-2 and 2-41-3. We do this by establishing a bijection between the avoiders and the trees. This bijection translates 7 statistics on the trees into statistics on the avoiders. \noindent Moreover, extensive computations suggest that the avoiders are structurally more closely connected to the beta(1,0)-trees---and thus to the planar maps---than two-stack sortable permutations are. In connection with this we give a nontrivial involution on the beta(1,0)-trees, which specializes to an involution on unlabeled rooted plane trees, where it yields interesting results.