The generalized Zagreb index for non-plane and plane recursive trees

The Zagreb index, which is defined as the sum of squares of degrees of the nodes of a tree, was studied in previous works by martingale techniques for random non-plane recursive trees and classes of random trees which are close to random plane recursive trees. These techniques are not easily amended to the generalized Zagreb index, which is defined similarly but with squares replaced by higher powers. We use the moment-transfer approach to (i) obtain the first-order asymptotics of moments and (ii) prove limit laws for the (suitably normalized) generalized Zagreb index for random non-plane and plane recursive trees. For the former, we show that for all higher powers the limit law is normal; for the latter, we show for cubes and fourth powers that it is a non-normal law.