Question

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