Several topological indces of random binary trees

The Zagreb index of random binary trees was mainly investigated under three standard probability models: random binary search trees, random Catalan trees and random digital search trees. Using an alternative method to study the number of leaves, the exact mean and variance of the Zagreb index of a random binary search tree was first obtained. After that, the asymptotic normality was derived by the contraction method. Then, the asymptotic normality for the Zagreb index of a random Catalan tree and a digital search tree were also given. Finally, two other topological indices named the Gordon-Scantlebury and Piatt indices, both closely related to the Zagreb index, were discussed in passing.