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 $\frac13t+\frac14s+\frac32$ leaves. We present an infinite series of graphs showing that our bound is tight.