  Given a vertex-weighted connected graph $G = (V, E)$, the maximum weight
internal spanning tree (MwIST for short) problem asks for a spanning tree $T$
of $G$ such that the total weight of the internal vertices in $T$ is maximized.
The un-weighted variant, denoted as MIST, is NP-hard and APX-hard, and the
currently best approximation algorithm has a proven performance ratio $13/17$.
The currently best approximation algorithm for MwIST only has a performance
ratio $1/3 - \epsilon$, for any $\epsilon > 0$. In this paper, we present a
simple algorithm based on a novel relationship between MwIST and the maximum
weight matching, and show that it achieves a better approximation ratio of
$1/2$. When restricted to claw-free graphs, a special case been previously
studied, we design a $7/12$-approximation algorithm.
