Number of incongruental unlabeled undirected trees with n nodes on a square lattice and edges of length 1 or sqrt(2) admitted to the 4 nearest or 4 2nd nearest neighbors.

as the only a(2)=2 candidates. Trees contributing to n=7 are

o.o-o

|\.\.

o.o-o

...\.

....o

or

o....

|\...

o.o-o

...X.

..o.o

./...

o....

where dashes are edges in E, NE, N, NW, W, SW, S or SE direction that connect nodes marked 'o' horizontally, vertically or along diagonals, and X's are crossing diagonal edges.