  Minimum $k$-Section denotes the NP-hard problem to partition the vertex set
of a graph into $k$ sets of sizes as equal as possible while minimizing the cut
width, which is the number of edges between these sets. When $k$ is an input
parameter and $n$ denotes the number of vertices, it is NP-hard to approximate
the width of a minimum $k$-section within a factor of $n^c$ for any $c<1$, even
when restricted to trees with constant diameter. Here, we show that every tree
$T$ allows a $k$-section of width at most $(k-1) (2 + 16n / diam(T) )
\Delta(T)$. This implies a polynomial-time constant-factor approximation for
the Minimum $k$-Section Problem when restricted to trees with linear diameter
and constant maximum degree. Moreover, we extend our results from trees to
arbitrary graphs with a given tree decomposition.
