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The asymptotic values of the general Zagreb and Randić indices of trees with bounded maximum degree

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. We show that the number of vertices of degree $j$ in $\mathcal {T}^Δ_n$ is asymptotically normal with mean $(μ_j+o(1))n$ and variance $(σ_j+o(1))n$, where $μ_j$, $σ_j$ are some constants. As a consequence, we give estimate to the value of the general Zagreb index for almost all trees in $\mathcal {T}^Δ_n$. Moreover, we obtain that the number of edges of type $(i,j)$ in $\mathcal {T}^Δ_n$ also has mean $(μ_{ij}+o(1))n$ and variance $(σ_{ij}+o(1))n$, where an edge of type $(i,j)$ means that the edge has one end of degree $i$ and the other of degree $j$, and $μ_{ij}$, $σ_{ij}$ are some constants. Then, we give estimate to the value of the general Randić index for almost all trees in $\mathcal {T}^Δ_n$.