Spanning trees with many leaves: new lower bounds in terms of number of vertices of degree 3 and at least 4

We prove that every connected graph with $s$ vertices of degree 3 and $t$ vertices of degree at least 4 has a spanning tree with $\frac25t+\frac15s+\alpha$ leaves, where $\alpha\ge\frac85$. Moreover, $\alpha\ge2$ for all graphs besides three exclusions. All exclusion are regular graphs of degree 4, they are explicitly described in the paper.
We present an infinite series of graphs, containing only vertices of degrees 3 and 4, for which the maximal number of leaves in a spanning tree is equal for $\frac25t+\frac15s+2$. Therefore we prove that our bound is tight.