On the number of spanning trees in labeled cactus

Let $t(Ca_n(n_2,n_3,\ldots))$ be a number of spanning trees in a labelled cactus with $n$ vertices, $n_2$ be a number of its edge-blocks, $n_2\ge0$, $n_i$ be a number of its polygon-blocks with $i$ vertices, $n_i\ge0$, $i\ge3$, and $k$ be a number of cycles in this cactus. We deduce the formula $t(Ca_n(n_2,n_3,\dots))=\prod_{i\ge3}i^{n_i}$, $n\ge2$, where $n-1=n_2+2n_3+\dots$ As consequence, we obtain inequalities $t(Ca_n(n_2,n_3,\dots))\le(\frac1k(n+k-n_2-1))^k\le(\frac1k(n+k-1))^k\le e^{n-1}$.