Leaf-to-leaf paths of many lengths

We prove that every tree of maximum degree $\mathrm{\Delta }$ with $\mathrm{\ell }$ leaves contains paths between leaves of at least ${\log }_{\mathrm{\Delta }-1}((\mathrm{\Delta }-2)\mathrm{\ell })$ distinct lengths. This settles in a strong form a conjecture of Narins, Pokrovskiy and Szabó. We also make progress towards another conjecture of the same authors, by proving that every tree with no vertex of degree 2 and diameter at least $N$ contains $N^{2\mathrm{/}3}\mathrm{/}6$ distinct leaf-to-leaf path lengths between $0$ and $N$.