  An ultrametric Cantor set can be seen as the boundary of a rooted weighted
tree called the Michon tree. The notion of Assouad dimension is re-interpreted
as seen on the Michon tree. The Assouad dimension of an ultrametric Cantor set
is finite if and only if the space is bi-Lipschitz embeddable in a finite
dimensional Euclidean space. This result, due to Assouad and refined by
Luukkainen--Movahedi-Lankarani is re-proved in the Michon tree formalism. It is
applied to answer the embedding question for some spaces which can be seen
naturally as boundary of trees: linearly repetitive subshifts, Sturmian
subshifts, and the boundary of Galton--Watson trees with random weights. Some
of these give examples of nonembeddable spaces with finite Hausdorff dimension.
