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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 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