In this paper we study the structure of suffix trees. Given an unlabeled tree $τ$ on $n$ nodes and suffix links of its internal nodes, we ask the question "Is $τ$ a suffix tree?", i.e., is there a string $S$ whose suffix tree has the same topological structure as $τ$? We place no restrictions on $S$, in particular we do not require that $S$ ends with a unique symbol. This corresponds to considering the more general definition of implicit or extended suffix trees. Such general suffix trees have many applications and are for example needed to allow efficient updates when suffix trees are built online. We prove that $τ$ is a suffix tree if and only if it is realized by a string $S$ of length $n-1$, and we give a linear-time algorithm for inferring $S$ when the first letter on each edge is known. This generalizes the work of I et al. [Discrete Appl. Math. 163, 2014].