  The Perron value $\rho(T)$ of a rooted tree $T$ has a central role in the
study of the algebraic connectivity and characteristic set, and it can be
considered a weight of spectral nature for $T$. A different, combinatorial
weight notion for $T$ - the moment $\mu(T)$ - emerges from the analysis of
Kemeny's constant in the context of random walks on graphs. In the present
work, we compare these two weight concepts showing that $\mu(T)$ is "almost" an
upper bound for $\rho(T)$ and the ratio $\mu(T)/\rho(T)$ is unbounded but at
most linear in the order of $T$. To achieve these primary goals, we introduce
two new objects associated with $T$ - the Perron entropy and the neckbottle
matrix - and we investigate how different operations on the set of rooted trees
affect the Perron value and the moment.
