2 Used a different triangle to represent the data.

I think this is what you want:

OEIS: the triangle of trees with n nodes and k leaves

(You should draw the sequence as a triangle to get the 2-dimensional information.)

                                            1
1       1   1   1   0
1       1       0
1       1       1       0
1       2       2       1       0
1       1   2   3       4       5   7   8   10  12
2       1       0
1       2   4       8       14  23  36  54  78
0   1   2   6       14  32  64  123 219
0   1   3       9   26  66  158 350
1       0
1       5       14      14      9       3       12  39  119 325
1       0
1       7       23      32      26      12      4       16  60  202
1       0
1       8       36      64      66      39      16      4       20  83
1       0
1       10      54      123         158         119         60      20      5       25
1       0
1       12      78      219         350         325         202         83      25      5       0   1       0


Edit: I edited to use a different representation of the data. I assume that the n-th row, k-th entry means the number of trees with n nodes and k leaves. See these other displays

1

I think this is what you want:

OEIS: the triangle of trees with n nodes and k leaves

(You should draw the sequence as a triangle to get the 2-dimensional information.)

1   1   1   1   1   1   1   1   1   1   1   1
0   1   1   2   3   4   5   7   8   10  12
0   1   2   4   8   14  23  36  54  78
0   1   2   6   14  32  64  123 219
0   1   3   9   26  66  158 350
0   1   3   12  39  119 325
0   1   4   16  60  202
0   1   4   20  83
0   1   5   25
0   1   5
0   1
0