On the number of spanning trees in labeled cactus

Let t(Ca_n(n₂,n₃,...)) be a number of spanning trees in a labelled cactus with n vertices, n₂ be a number of its edge-blocks, n₂ ^ 0, n_i be a number of its polygon-blocks with i vertices, n_i ^ 0, i ^ 3, and k be a number of cycles in this cactus. We deduce the formula t(Ca_n(n₂, n₃, ...)) = П iⁿⁱ, n ^ 2, where n - 1 = n₂ + 2n₃ +... As consequence, we ob- ⁽₁ V ⁽₁ V tain inequalities t(Ca_n(n₂ ,n₃,...)) -(n + k - n₂ - 1) j -(n + k - 1) j ^ eⁿ 1.

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