A maximally graceful tree is a graceful tree (or assuming the truth of the graceful tree theorem, simply "a tree") that has the maximum possible number of graceful labelings among all trees with the same number of vertices.
Anick (2016) enumerated maximally graceful trees on up to 17 vertices (16 edges), including both subtractively complementary labelings in his counts (meaning they are a factor of 2 times the usual "fundamentally distinct" labeling counts).
The numbers of fundamentally distinct labelings of a maximally graceful tree on 
, 2, ... vertices (corresponding to
,
1, 2, ... edges) are 1, 1, 1, 1, 3, 6, 18, 52, 114, 367, 1777, 5249, 21107, 84746,
432769, 10399350, ... (OEIS A379102). The corresponding
maximally labeled trees are illustrated above, with two distinct trees tied for the
maximum in the cases of 
and 
.
