Any spanning tree of the n x n grid has fundamental cycles with average length Omega(log n), and the bound is tight.
Hlinˇ en´ y:Branch-width, parse trees, and monadic second-order logic for matroids, Journal of Combinatorial Theory Series B96(2006), 325– 351
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Fundamental cycles in grid graphs
Any spanning tree of the n x n grid has fundamental cycles with average length Omega(log n), and the bound is tight.