  For the simple Bratteli diagrams B where there is a single edge connecting
any two vertices in consecutive levels, we show that a random order has
uncountably many infinite paths if and only if the growth rate of the level-n
vertex sets is super-linear. This gives us the dichotomy: a random order on a
slowly growing Bratteli diagram admits a homeomorphism, while a random order on
a quickly growing Bratteli diagram does not. We also show that for a large
family of infinite rank Bratteli diagrams, a random order on B does not admit a
continuous Vershik map.
