The asymptotic values of the general Zagreb and Randi\'c indices of trees with bounded maximum degree

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. We show that the number of vertices of degree $j$ in $\mathcal {T}^{\Delta}_n$ is asymptotically normal with mean $(\mu_j+o(1))n$ and variance $(\sigma_j+o(1))n$, where $\mu_j$, $\sigma_j$ are some constants. As a consequence, we give estimate to the value of the general Zagreb index for almost all trees in $\mathcal {T}^{\Delta}_n$. Moreover, we obtain that the number of edges of type $(i,j)$ in $\mathcal {T}^{\Delta}_n$ also has mean $(\mu_{ij}+o(1))n$ and variance $(\sigma_{ij}+o(1))n$, where an edge of type $(i,j)$ means that the edge has one end of degree $i$ and the other of degree $j$, and $\mu_{ij}$, $\sigma_{ij}$ are some constants. Then, we give estimate to the value of the general Randi\'{c} index for almost all trees in $\mathcal {T}^{\Delta}_n$.