In the Distance-constrained Vehicle Routing Problem (DVRP), we are given a graph with integer edge weights, a depot, a set of $n$ terminals, and a distance constraint $D$. The goal is to find a minimum number of tours starting and ending at the depot such that those tours together cover all the terminals and the length of each tour is at most $D$. The DVRP on trees is of independent interest, because it is equivalent to the virtual machine packing problem on trees studied by Sindelar et al. [SPAA'11]. We design a simple and natural approximation algorithm for the tree DVRP, parameterized by $\varepsilon >0$. We show that its approximation ratio is $α+ \varepsilon$, where $α\approx 1.691$, and in addition, that our analysis is essentially tight. The running time is polynomial in $n$ and $D$. The approximation ratio improves on the ratio of 2 due to Nagarajan and Ravi [Networks'12]. The main novelty of this paper lies in the analysis of the algorithm. It relies on a reduction from the tree DVRP to the bounded space online bin packing problem via a new notion of reduced length.