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

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