We study the problem of low-stretch spanning trees in graphs of bounded
width: bandwidth, cutwidth, and treewidth. We show that any simple connected
graph G with a linear arrangement of bandwidth b can be embedded into a
distribution T of spanning trees such that the expected stretch of
each edge of G is O(b2). Our proof implies a linear time algorithm for
sampling from T. Therefore, we have a linear time algorithm that
finds a spanning tree of G with average stretch O(b2) with high
probability. We also describe a deterministic linear-time algorithm for
computing a spanning tree of G with average stretch O(b3). For graphs of
cutwidth c, we construct a spanning tree with stretch O(c2) in linear
time. Finally, when G has treewidth k we provide a dynamic programming
algorithm computing a minimum stretch spanning tree of G that runs in
polynomial time with respect to the number of vertices of G