Reconstructing Arbitrary Trees from Traces in the Tree Edit Distance Model

In this paper, we consider the problem of reconstructing trees from traces in the tree edit distance model. Previous work by Davies et al. (2019) analyzed special cases of reconstructing labeled trees. In this work, we significantly expand our understanding of this problem by giving general results in the case of arbitrary trees. Namely, we give: a reduction from the tree trace reconstruction problem to the more classical string reconstruction problem when the tree topology is known, a lower bound for learning arbitrary tree topologies, and a general algorithm for learning the topology of any tree using techniques of Nazarov and Peres (2017). We conclude by discussing why arbitrary trees require exponentially many samples under the left propagation model.