We investigate a logic of an algebra of trees including the update operation, which expresses that a tree is obtained from an input tree by replacing a particular direct subtree of the input tree, while leaving the rest intact. This operation improves on the expressivity of existing logics of tree algebras in our case of feature trees, which allow for an unbounded number of children of a node in a tree.

We show that the first-order theory of this algebra is decidable via a weak quantifier elimination procedure which is allowed to swap existential quantifiers for universal quantifiers. This study is motivated by the logical modeling of transformations on UNIX file system trees expressed in a simple programming language.