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

Let ${\mathcal{T}}_n^{\mathrm{\Delta }}$ denote the set of trees of order $n$, in which the degree of each vertex is bounded by some integer $\mathrm{\Delta }$. Suppose that every tree in ${\mathcal{T}}_n^{\mathrm{\Delta }}$ is equally likely. For any given subtree $H$, we show that the number of occurrences of $H$ in trees of ${\mathcal{T}}_n^{\mathrm{\Delta }}$ 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 ${\mathcal{T}}_n^{\mathrm{\Delta }}$, and give an explanation in theory to the approximate linear correlation between $EE$ and the first Zagreb index obtained by quantitative analysis.