  For an integer $k$, a $k$-tree is a tree with maximum degree at most $k$.
More generally, if $f$ is an integer-valued function on vertices, an $f$-tree
is a tree in which each vertex $v$ has degree at most $f(v)$. Let $c(G)$ denote
the number of components of a graph $G$. We show that if $G$ is a connected
$K_4$-minor-free graph and
  $$
  c(G-S) \;\le\; \sum_{v \in S} (f(v)-1)
  \quad\hbox{for all $S \subseteq V(G)$ with $S \ne \emptyset$}
  $$ then $G$ has a spanning $f$-tree. Consequently, if $G$ is a
$\frac{1}{k-1}$-tough $K_4$-minor-free graph, then $G$ has a spanning $k$-tree.
These results are stronger than results for general graphs due to Win (for
$k$-trees) and Ellingham, Nam and Voss (for $f$-trees). The $K_4$-minor-free
graphs form a subclass of planar graphs, and are identical to graphs of
treewidth at most $2$, and also to graphs whose blocks are series-parallel. We
provide examples to show that the inequality above cannot be relaxed by adding
$1$ to the right-hand side, and also to show that our result does not hold for
general planar graphs. Our proof uses a technique where we incorporate
toughness-related information into weights associated with vertices and
cutsets.
