The passage quoted below is from "The number of spanning trees of a graph" by Jianxi Li, Wai Chee Shiu, and An Chang, Applied Mathematics Letters 23.3 (2010): 286-290, DOI:10.1016/j.aml.2009.10.006, MR2565192, Zbl 1227.05125.

Could someone clearly explain the proof of the inequality on the number of spanning trees of a graph $G$ in this passage?

This inequality seems to contradict Cayley's formula for the number of spanning trees of the complete graph. I tested a complete graph $G=K_5$. This formula gave me $121$ as an upper bound for the number of spanning trees while Cayley's gave me $125$. I am not able to figure out how it went wrong.

Could someone provide some insights?