  The Virasoro algebra determines all `graviton' matrix elements in
AdS$_3$/CFT$_2$. We study the explicit exchange of any number of Virasoro
gravitons between heavy and light CFT$_2$ operators at large central charge.
These graviton exchanges can be written in terms of new on-shell tree diagrams,
organized in a perturbative expansion in $h_H/c$, the heavy operator dimension
divided by the central charge. The Virasoro vacuum conformal block, which is
the sum of all the tree diagrams, obeys a differential recursion relation
generalizing that of the Catalan numbers. We use this recursion relation to sum
the on-shell diagrams to all orders, computing the Virasoro vacuum block.
Extrapolating to large $h_H/c$ determines the Hawking temperature of a BTZ
black hole in dual AdS$_3$ theories.
