If the vertices of a graph are labeled 1 to n, the graph is graceful
if every edge u→v corresponds to a unique difference abs(u-v).
Since there are only n-1 possible differences, the graph contains at most n-1 edges.
A connected graceful graph is a tree.
An unlabeled graph is graceful if it can be labeled gracefully.
Connect the point called 1 to all the other points,
and you have a graceful graph.
Hence a graceful tree exists for every n.