  We study the total $\alpha$-powered length of the rooted edges in a random
minimal directed spanning tree - first introduced in Bhatt and Roy (2004) - on
a Poisson process with intensity $s \ge 1$ on the unit cube $[0,1]^d$ for $d
\ge 3$. While a Dickman limit was proved in Penrose and Wade (2004) in the case
of $d=2$, in dimensions three and higher, Bai, Lee and Penrose (2006) showed a
Gaussian central limit theorem when $\alpha=1$, with a rate of convergence of
the order $(\log s)^{-(d-2)/4} (\log \log s)^{(d+1)/2}$. In this paper, we
extend these results and prove a central limit theorem in any dimension $d \ge
3$ for any $\alpha>0$. Moreover, making use of recent results in Stein's method
for region-stabilizing functionals, we provide presumably optimal
non-asymptotic bounds of the order $(\log s)^{-(d-2)/2}$ on the Wasserstein and
the Kolmogorov distances between the distribution of the total $\alpha$-powered
length of rooted edges, suitably normalized, and that of a standard Gaussian
random variable.
