On spanning trees with high internal degree
  Alon and Wormald showed that any graph with minimum degree d contains a spanning star forest in which every connected component is of size at least \Omega((d/\log d)^{1/3}). They asked if any connected graph with minimum degree at least d has a spanning tree in which every internal vertex has degree at least cd/\log d, for some absolute constant c > 0.   We give a simple example showing that this is not the case. 
Codrut Grosu
math.CO