Some results on spanning trees.

Some structures of spanning trees with many or less leaves in a connected graph are determined. We show (1) a connected graph G has a spanning tree T with minimum leaves such that T contains a longest path, and (2) a connected graph G on n vertices contains a spanning tree T with the maximum leaves such that Δ( G) = Δ( T) and the number of leaves of T is not greater than n − D( G) + 1, where D( G) is the diameter of G. [ABSTRACT FROM AUTHOR]