Universal quasiconformal trees

A quasiconformal tree is a doubling (compact) metric tree in which the diameter of each arc is comparable to the distance of its endpoints. We show that for each integer $n\ge 2$, the class of all quasiconformal trees with uniform branch separation and valence at most $n$, contains a quasisymmetrically ‘‘universal’’ element, that is, an element of this class into which every other element can be embedded quasisymmetrically. We also show that every quasiconformal tree with uniform branch separation quasisymmetrically embeds into ${\mathbb{R}}^2$. Our results answer two questions of Bonk and Meyer from 2022, in higher generality, and partially answer one question of Bonk and Meyer from 2020.