Trees with many leaves in tournaments

Sumner's universal tournament conjecture states that every $(2n-2)$-vertex tournament should contain a copy of every $n$-vertex oriented tree. If we know the number of leaves of an oriented tree, or its maximum degree, can we guarantee a copy of the tree with fewer vertices in the tournament? Due to work initiated by H\"aggkvist and Thomason (for number of leaves) and K\"uhn, Mycroft and Osthus (for maximum degree), it is known that improvements can be made over Sumner's conjecture in some cases, and indeed sometimes an $(n+o(n))$-vertex tournament may be sufficient. In this paper, we give new results on these problems. Specifically, we show i) for every $\alpha>0$, there exists $n_0\in\mathbb{N}$ such that, whenever $n\geqslant n_0$, every $((1+\alpha)n+k)$-vertex tournament contains a copy of every $n$-vertex oriented tree with $k$ leaves, and ii) for every $\alpha>0$, there exists $c>0$ and $n_0\in\mathbb{N}$ such that, whenever $n\geqslant n_0$, every $(1+\alpha)n$-vertex tournament contains a copy of every $n$-vertex oriented tree with maximum degree $\Delta(T)\leqslant cn$. Our first result gives an asymptotic form of a conjecture by Havet and Thomass\'e, while the second improves a result of Mycroft and Naia which applies to trees with polylogarithmic maximum degree.