The asymptotic behavior of the Estrada index for trees

Let T?n denote the set of trees of order n, in which the degree of each vertex is bounded by some integer ?. Suppose that every tree in T?n is equally likely. For any given small tree H, we first show that the number of occurrences of H in trees of T?n has mean (?H+o(1))n and variance (?H+o(1))n, where ?H, ?H are some constants. Then we apply this result to estimate the value of the Estrada index EE for almost all trees in T?n, and give a theoretical explanation to the approximate linear correlation between EE and the first Zagreb index obtained by quantitative analysis.