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Let $\mathcal {T}^{\Delta}_n$ denote the set of trees of order $n$, in whichthe degree of each vertex is bounded by some integer $\Delta$. Suppose thatevery tree in $\mathcal {T}^{\Delta}_n$ is equally likely. For any givensubtree $H$, 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 anapplication, we estimate the value of the Estrada index $EE$ for almost alltrees in $\mathcal {T}^{\Delta}_n$, and give an explanation in theory to theapproximate linear correlation between $EE$ and the first Zagreb index obtainedby quantitative analysis.

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