A compacted binary tree is a directed acyclic graph encoding a binary tree in which common subtrees are factored and shared, such that they are represented only once. We show that the number of compacted binary trees of size n grows asymptotically like where is the largest root of the Airy function. Our method involves a new two parameter recurrence which yields an algorithm of quadratic arithmetic complexity for computing the number of compacted trees up to a given size. We use empirical methods to estimate the values of all terms defined by the recurrence, then we prove by induction that these estimates are sufficiently accurate for large n to determine the asymptotic form. Our results also lead to new bounds on the number of minimal finite automata recognizing a finite language on a binary alphabet. As a consequence, these also exhibit a stretched exponential.