Abstract
We disprove the conjecture that every tree with 𝑡 leaves embeds isometrically into ℓ∞[log2𝑡]. We construct a 32-leaf tree whose least isometric ℓ∞-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.