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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 as below to get the 2-dimensional information.)

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