Two graphs G and H are hypomorphic if there exists a bijection φ:V(G)→V(H) such that G-v≅H-φ(v) for each vϵV(G). A graph G is reconstructible if H G for all H hypomorphic to G. It is well known that not all infinite graphs are reconstructible. However, the Harary-Schwenk-Scott Conjecture from 1972 suggests that all locally finite trees are reconstructible. In this paper, we construct a counterexample to the Harary-Schwenk-Scott Conjecture. Our example also answers four other questions of Nash-Williams, Halin and Andreae on the reconstruction of infinite graphs.