  Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist
highly connected graphs in which every spanning tree contains vertices of
degree 2. Using a result of Alon and Wormald, we show that there exists a
natural number $d$ such that every graph of minimum degree at least $d$
contains a spanning tree without adjacent vertices of degree 2. Moreover, we
prove that every graph with minimum degree at least 3 has a spanning tree
without three consecutive vertices of degree 2.
