In this paper we consider recovering combinatorial objects from many noisy observations. The first part of the paper concerns reconstructing trees from traces in the tree edit distance model. Previous work focused on reconstructing various classes of labelled trees, while our work gives reductions from the classic string reconstruction setting to unlabelled trees. In the second part of the paper we discuss combinatorial identities of the binary deletion channel on finite and infinite strings. We link probabilities of observing bits in a trace to derivatives of certain generating functions. We also give identities for the deletion channel, conditioned on traces having fixed length.