Abstract
A digraph is n-unavoidable if it is contained in every tournament of order n. We first prove that every arborescence of order n with k leaves is (n+k−1)-unavoidable. We then prove that every oriented tree of order n (n≥2) with k leaves is (32n+32k−2)-unavoidable and (92n−52k−92)-unavoidable, and thus (218n−4716)-unavoidable. Finally, we prove that every oriented tree of order n with k leaves is (n+144k2−280k+124)-unavoidable.
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