Starting from combinatorial structures, one can study some of their characteristics by means of attribute grammars [1, 2]. This leads to multivariate generating functions that permit us to study the distribution of these characteristics, part of it automatically.
F_{i}(B)= 

f 
æ ç ç è 
d_{i}^{m}+ 

a_{i,j}^{m}F_{j}  (  B_{k}^{m}  ) 
ö ÷ ÷ ø 
+g_{i} 
B(z  )= 

z 

G 

æ è 
z 

B_{k}^{m}(z 

) 
ö ø 
B(z  )= 

z 

... z 

. 
B(z)=z 


z 



æ ç ç è 

z 

ö ÷ ÷ ø 

. 
B(z)=z 

G 

æ è 
z 

C(z 

) 
ö ø 
. 
To describe increasing trees with attribute grammars, we need to introduce the Greene operator also called box operator [4]. In a labelled structure, the Greene operator specifies where the minimum label is to be. For example the increasing trees are defined by T=e T_{1}·Min(N)· T_{2} which specifies that the minimum is in the root N. The generating function has been determined by Greene:
T(z)=  ó õ 

T^{2}(x) 

dx. 
T(z,u)=1+  ó õ 

æ ç ç è 

x 
ö ÷ ÷ ø 
T(xu,u)^{2} dx. 

=2H_{n}3+ 

with T(z,1)= 

F(A)=Union(b_1*F_1(B)+...+b_k*F_k(B),c_1*F_1(C)+...+c_k*F_k(C)) + a_1*F_1(A)+...+a_k*F_k(A)+a_0.
This document was translated from L^{A}T_{E}X by H^{E}V^{E}A.