Generating trees and proper Riordan Arrays

We use an algebraic approach to study the connection between generating trees and proper Riordan Arrays deriving a theorem that, under suitable conditions, associates a Riordan Array to a generating tree and vice versa. Thus, we can use results from the theory of Riordan Arrays to study properties of generating trees. In particular, we can find, in a general and easy way, the generating functions counting the distribution of trees' nodes at each level. The connection between Riordan Arrays and transfer matrices is also shown.