 Bradley-Terry maximum-likelihood estimate on a comparison tree

For a connected tree of comparisons with every observed win fraction strictly between zero and one, the Bradley-Terry model can fit every empirical edge probability exactly. Fix one positive strength to remove scaling ambiguity, then propagate strength ratios $p/(1-p)$ along tree edges. Edge log-ratios are independent real coordinates and each binomial log-likelihood term is strictly concave, proving existence and uniqueness of the normalized estimate. Cycles would impose extra compatibility relations.