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The leaf distance of a tree is the maximum d such that the distance between any pair of leaves in the tree is at least d . Kaneko provided sufficient conditions to force the existence of a spanning tree with leaf distance at least d = 3 and conjectured that similar conditions suffice for larger d . The case when d = 4 was later proved by Kaneko, Kano, and Suzuki. In this paper, we show that when d ≥ 4 , a stronger form of this conjecture holds for graphs with independence number at most five. As an immediate corollary, we obtain that when d ≥ n ∕ 3 , this stronger version holds for all n -vertex graphs, consequently proving Kaneko’s conjecture for d ≥ n ∕ 3 . [ABSTRACT FROM AUTHOR]