Motivated by the spread of on-line information in general and on-line petitions in particular, recent research has raised the following combinatorial estimation problem. There is a tree T that we cannot observe directly (representing the structure along which the information has spread), and certain nodes randomly decide to make their copy of the information public. In the case of a petition, the list of names on each public copy of the petition also reveals a path leading back to the root of the tree. What can we conclude about the properties of the tree we observe from these revealed paths, and can we use the structure of the observed tree to estimate the size of the full unobserved tree T? Here we provide the first algorithm for this size estimation task, together with provable guarantees on its performance. We also establish structural properties of the observed tree, providing the first rigorous explanation for some of the unusual structural phenomena present in the spread of real chain-letter petitions on the Internet.