Finite height lamination spaces for surfaces

We describe spaces of essential finite height (measured) laminations in a surface $S$ using a parameter space we call $\mathbb S$, an ordered semi-ring. We show that for every finite height essential lamination $L$ in $S$, there is an action of $\pi_1(S)$ on an $\mathbb S$-tree dual to the lift of $L$ to the universal cover of $S$.