Max-linear Bayesian networks are recursive max-linear structural equation models represented by an edge weighted directed acyclic graph (DAG). The identifiability and estimation of max-linear Bayesian networks is an intricate issue as Gissibl, Klüppelberg, and Lauritzen have shown. As such, a max-linear Bayesian network is generally unidentifiable and standard likelihood theory cannot be applied. We can associate tropical polyhedra to max-linear Bayesian networks. Using this, we investigate the minimum-ratio estimator proposed by Gissibl, Klüppelberg, and Lauritzen and give insight on the structure of minimal best-case samples for parameter recovery which we describe in terms of set covers of certain triangulations. We also combine previous work on estimating max-linear models from Tran, Buck, and Klüppelberg to apply our geometric approach to the structural inference of max-linear models. This is tested extensively on simulated data and on real world data set, the NHANES report for 2015--2016 and the upper Danube network data.