There exists a bijection between one-stack sortable permutations (permutations which avoid the pattern (231)) and rooted plane trees. We define an edit distance between permutations which is consistent with the standard edit distance between trees. This one-to-one correspondence yields a polynomial algorithm for the subpermutation problem for (231) pattern-avoiding permutations.Moreover, we obtain the generating function of the edit distance between ordered unlabeled trees and some special ones.For the general case we show that the mean edit distance between a rooted plane tree and all other rooted plane trees is at least n/ln(n).Some results can be extended to labeled trees considering coloredDyck paths or, equivalently, colored one-stack sortable permutations.

Word and tree codes are studied in a common framework, that of polypodes which are sets endowed with a substitution like operation. Many examples are given and basic properties are examined. The code decomposition theorem is valid in this general setup.

We give a new characterization of the hyperarithmetic sets: a set X of integers is recursive in e α if and only if there is a Turing machine which computes X and “halts” in less than or equal to the ordinal number ωα of steps. This result represents a generalization of the well-known “limit lemma” due to J. R. Shoenfield [Sho-1] and later independently by H. Putnam [Pu] and independently by E. M. Gold [Go]. As an application of this result, we give a recursion theoretic analysis of clopen determinacy: there is a correlation given between the height α of a well-founded tree corresponding to a clopen game A ⊆ ωω and the Turing degree of a winning strategy ƒ for one of the players—roughly, ƒ can be chosen to be recursive in 0 α and this is the best possible (see §4 for precise results).