Metrics on trees I. The tower algorithm for interval maps

We consider Milnor's "tower algorithm" in the space of piecewise monotone maps, an iterative algorithm on the space of metrics which unifies, on the one hand, Thurston's iterative scheme which converges to holomorphic models, and, on the other hand, the theory of piecewise linear models coming from kneading theory. We prove that the algorithm converges for unimodal maps of high entropy, and provide examples where it does not converge for unimodal maps of low entropy and multimodal maps of higher degree.