  Given a graph $G = (V, E)$, we wish to compute a spanning tree whose maximum
vertex degree, i.e. tree degree, is as small as possible. Computing the exact
optimal solution is known to be NP-hard, since it generalizes the Hamiltonian
path problem. For the approximation version of this problem, a $\tilde{O}(mn)$
time algorithm that computes a spanning tree of degree at most $\Delta^* +1$ is
previously known [F\"urer \& Raghavachari 1994]; here $\Delta^*$ denotes the
minimum tree degree of all the spanning trees. In this paper we give the first
near-linear time approximation algorithm for this problem. Specifically
speaking, we propose an $\tilde{O}(\frac{1}{\epsilon^7}m)$ time algorithm that
computes a spanning tree with tree degree $(1+\epsilon)\Delta^* +
O(\frac{1}{\epsilon^2}\log n)$ for any constant $\epsilon \in (0,\frac{1}{6})$.
Thus, when $\Delta^*=\omega(\log n)$, we can achieve approximate solutions with
constant approximate ratio arbitrarily close to 1 in near-linear time.
