Tree pressure for hyperbolic and non-exceptional upper semi-continuous potentials

In this note, we investigate the tree pressure for multi-modal interval maps with a certain class of hyperbolic and non-exceptional upper semi-continuous functions. In particular, we obtain a generalized version of Corollary 2.2 in the paper \cite{LRL14} by Li and Rivera-Letelier. This property will be used to prove the existence of a conformal measure for the geometric potential in the negative spectrum.