A new formulation of Lagrange inversion for several variables will be described which does not involve a determinant. This formulation is convenient for the asymptotic investigation of numbers defined by Lagrange inversion. Examples of tree problems where the number of vertices of degree k are counted and where vertices are 2colored will be given. Noncrossing partitions give another example and the MeirMoon formula for powers of an inversion is a special case.
Enumeration of such trees is best done by taking into account the colors of the vertices: let x_{1} and x_{2} mark the green and red vertices, and define w_{1}(x_{1},x_{2}) and w_{2} (x_{1},x_{2}) as the functions enumerating the trees whose root is green (resp. red). These functions satisfy the system of equations
[ x^{n} ] w(x) = 

[ t^{n1} ] f (t)^{n}; [ x^{n}] g(w(x)) = 

[ t^{n1} ] g'(t) f^{n} (t) . 
[ 


] g( 

( 

)) = [ 


] 
æ ç ç ç è 
g( 

) 


( 

) 
½ ½ ½ ½ ½ 
½ ½ ½ ½ ½ 
d_{i,j}  

½ ½ ½ ½ ½ 
½ ½ ½ ½ ½ 
ö ÷ ÷ ÷ ø 
, 
Now the derivative of a (d+1)dimensional function f according to such a tree is a product on (d+1) terms, where f_{i} is differentiated according to the incoming edges into the vertex labelled by i; this is best explained on the above example, with f = (f_{0},f_{1},f_{2}):^{1}

= 

· f_{1} · f_{2}; 

= 

· f_{1} · 

; 

= 

· 

· f_{2}. 
[ 


] g ( 

( 

)) = 
æ ç ç è 

n_{i} 
ö ÷ ÷ ø 

[ 


] 


, 






h:= 

+ n_{2} 



+ n_{1} 



. 
k_{1} = n_{1} t_{1} 


+ n_{2} t_{1} 


; k_{2} = n_{1} t_{2} 


+ n_{2} t_{2} 


. 
k_{1} = n_{1} 

+ n_{2} 

; k_{2} = n_{1} 

+ n_{2} 

. 
t_{1} = 

=: r_{1}; t_{2} = 

=: r_{2}. 
n 


. 
This document was translated from L^{A}T_{E}X by H^{E}V^{E}A.