  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 $\mathcal {T}^{\Delta}_n$.
