In the recent past, a special type of distance function, namely monometrics, has garnered a lot of attention because of its value and interest in both theory and applications. The key challenge lies in finding a monometric w.r.t. an arbitrary betweenness relation. To address this, the constructions of non-trivial monometrics on various types of betweenness relations have been explored. While it was shown that a non-trivial monometric may not always exist on arbitrary betweenness relations, the study of such betweenness relations has not been undertaken. In this paper, we take a step towards filling this gap by partially characterizing betweenness relations that will not give rise to non-trivial monometrics. By addressing this fundamental question, we hope to provide a clearer framework for understanding when monometrics can be defined.