  Given a connected vertex-weighted graph $G$, the maximum weight internal
spanning tree (MaxwIST) problem asks for a spanning tree of $G$ that maximizes
the total weight of internal nodes. This problem is NP-hard and APX-hard, with
the currently best known approximation factor $1/2$ (Chen et al., Algorithmica
2019). For the case of claw-free graphs, Chen et al. present an involved
approximation algorithm with approximation factor $7/12$. They asked whether it
is possible to improve these ratios, in particular for claw-free graphs and
cubic graphs.
  We improve the approximation factors for the MaxwIST problem in cubic graphs
and claw-free graphs. For cubic graphs we present an algorithm that computes a
spanning tree whose total weight of internal vertices is at least
$\frac{3}{4}-\frac{3}{n}$ times the total weight of all vertices, where $n$ is
the number of vertices of $G$. This ratio is almost tight for large values of
$n$. For claw-free graphs of degree at least three, we present an algorithm
that computes a spanning tree whose total internal weight is at least
$\frac{3}{5}-\frac{1}{n}$ times the total vertex weight. The degree constraint
is necessary as this ratio may not be achievable if we allow vertices of degree
less than three.
  With the above ratios, we immediately obtain better approximation algorithms
with factors $\frac{3}{4}-\epsilon$ and $\frac{3}{5}-\epsilon$ for the MaxwIST
problem in cubic graphs and claw-free graphs of degree at least three, for any
$\epsilon>0$. In addition to improving the approximation factors, the new
algorithms are relatively short compared to that of Chen et al.. The new
algorithms are fairly simple, and employ a variant of the depth-first search
algorithm that selects a relatively-large-weight vertex in every branching
step. Moreover, the new algorithms take linear time while previous algorithms
for similar problem instances are super-linear.
