Trees · Hard problem

Network $P$ has vertices $1$–$6$ and cables $12, 23, 34, 41, 45, 56, 63$ (two squares, $1\text{-}2\text{-}3\text{-}4$ and $3\text{-}4\text{-}5\text{-}6$, sharing edge $34$). Network $Q$ has the same $6$ vertices wired as a hexagon $12,23,34,45,56,61$ plus two long chords $14$ and $25$. Which network has MORE spanning trees, and how many does it have?