  We study two problems that seek a subtree $T$ of a graph $G=(V,E)$ such that
$T$ satisfies a certain property and has minimal maximum degree.
  - In the Min-Degree Group Steiner Tree problem we are given a collection
${\cal S}$ of groups (subsets of $V$) and $T$ should contain a node from every
group.
  - In the Min-Degree Steiner $k$-Tree problem we are given a set $R$ of
terminals and an integer $k$, and $T$ should contain at least $k$ terminals.
  We show that if the former problem admits approximation ratio $\rho$ then the
later problem admits approximation ratio $\rho \cdot O(\log k)$. For bounded
treewidth graphs, we obtain approximation ratio $O(\log^3 n)$ for Min-Degree
Group Steiner Tree.
  In the more general Bounded Degree Group Steiner Tree problem we are also
given edge costs and degree bounds $\{b(v):v \in V\}$, and $T$ should obey the
degree constraints $deg_T(v) \leq b(v)$ for all $v \in V$. We give a bicriteria
$(O(\log N \log |{\cal S}|),O(\log^2 n))$-approximation algorithm for this
problem on tree inputs, where $N$ is the size of the largest group,
generalizing the approximation of Garg, Konjevod, and Ravi for the case without
degree bounds.
