Trees with three leaves are (n+1)-unavoidable Original Research Article

We prove that every tree of order n⩾5 with three leaves is (n+1)-unavoidable. More precisely, we prove that every tree A with three leaves of order n is contained in every tournament T of order n+1 except if (T;A) is (R5;S3+) or its dual, where R5 is the regular tournament on five vertices and S3+ is the outstar of degree three, i.e. the tree consisting of a root dominating three leaves. We then deduce that Sumnerʹs conjecture is true for trees with four leaves, i.e. every tree of order n with four leaves is (2n−2)-unavoidable.