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* <math> p_{i,i}\leq p_{i,k} + p_{k,j} </math> for all <math> i,j,k\in [1,n]: i\neq j\neq k</math>
* <math> \sum_{j=1}^{n} 1/2^{p_{i,j}}=1/2 </math> for all <math> i\in [1,n]</math> (which is an adaptation of the [[Kraft–McMillan inequality]])
* <math> \sum_{i=1}^n\sum_{j=1}^{n} p_{i,j}/2^{p_{i,j}}=2n-3 </math>, also referred to as the ''phylogenetic manifold''<ref>{{cite journal | vauthors = Catanzaro D, Pesenti R, Wolsey L | title = On the Balanced Minimum Evolution Polytope | journal = Discrete Optimization | volume = 36 | pages = 1-33 | date = 2020 }}</ref>.
These equalities are proved to be necessary and independent for a path-length collection to encode an UBT with n leaves<ref>{{cite journal | vauthors = Catanzaro D, Pesenti R, Wolsey L | title = On the Balanced Minimum Evolution Polytope | journal = Discrete Optimization | volume = 36 | pages = 1-33 | date = 2020 }}</ref>. It is currently unknown whether they are also sufficient.
==Alternative names==
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