A 32-leaf tree requiring six coordinates for an isometric ${\mathrm{\ell }}_{\mathrm{\infty }}$ embedding

We disprove the conjecture that every tree with t leaves embeds isometrically into ${\mathrm{\ell }}_{\mathrm{\infty }}^{\lceil {\log }_{2}t\rceil }$. We construct a 32-leaf tree whose least isometric ${\mathrm{\ell }}_{\mathrm{\infty }}$-dimension is six rather than five, and prove that every tree with at most 31 leaves attains the conjectured bound; Brigham et al. had recorded equality through 21 leaves. Thus 32 is the first failure, and the example answers affirmatively a question of Fitzpatrick and Nowakowski from 2000. The same topology has dimension six under every assignment of positive edge lengths, and therefore also disproves the later sharp leaf-threshold conjecture for weighted metric trees.