The asymptotic number of occurrences of a subtree in trees with bounded maximum degree and an application to the Estrada index

Let $\mathcal {T}^{\Delta}_n$ denote the set of trees of order $n$, in which the degree of each vertex is bounded by some integer $\Delta$. Suppose that every tree in $\mathcal {T}^{\Delta}_n$ is equally likely. For any given subtree $H$, we show that the number of occurrences of $H$ in trees of $\mathcal {T}^{\Delta}_n$ is with mean $(\mu_H+o(1))n$ and variance $(\sigma_H+o(1))n$, where $\mu_H$, $\sigma_H$ are some constants. As an application, we estimate the value of the Estrada index $EE$ for almost all trees in $\mathcal {T}^{\Delta}_n$, and give an explanation in theory to the approximate linear correlation between $EE$ and the first Zagreb index obtained by quantitative analysis.