中文摘要：

主要研究了随机二叉搜索树、随机Catalan树、随机数字搜索树这3种随机二叉树模型下的Zagreb指数．首先得到随机二叉搜索树关于该指数的均值和方差，再用一种新的方法即压缩法得到渐近正态性．然后给出了随机Catalan树和随机数字搜索树的同种指数的渐近正态性．最后简单讨论了两个与Zagreb指数密切相关的拓扑指数即GordowScantlebury和Platt指数．

英文摘要：

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 Platt indices, both closely related to the Zagreb index, were discussed in passing.