  The edge-reconstruction number ern$(G)$ of a graph $G$ is equal to the
minimum number of edge-deleted subgraphs $G-e$ of $G$ which are sufficient to
determine $G$ up to isomorphsim. Building upon the work of Molina and using
results from computer searches by Rivshin and more recent ones which we carried
out, we show that, apart from three known exceptions, all bicentroidal trees
have edge-reconstruction number equal to 2. We also exhibit the known trees
having edge-reconstruction number equal to 3 and we conjecture that the three
infinite families of unicentroidal trees which we have found to have
edge-reconstruction number equal to 3 are the only ones.
