Spanning trees with many leaves: lower bounds in terms of number of   vertices of degree 1, 3 and at least~4
  We prove that every connected graph with $s$ vertices of degree~1 and 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${1\over 3}t +{1\over 4}s+{3\over 2}$ leaves. We present infinite series of graphs showing that our bound is tight. 
Dmitri Karpov
math.CO