  A {\it uniformly $p$-to-one endomorphism} is a measure-preserving map with
entropy log $p$ which is almost everywhere $p$-to-one and for which the
conditional expectation of each preimage is precisely $1/p$. The {\it standard}
example of this is a one-sided $p$-shift with uniform i.i.d. Bernoulli measure.
We give a characterization of those uniformly finite-to-one endomorphisms
conjugate to this standard example by a condition on the past tree of names
which is analogous to {\it very weakly Bernoulli} or {\it loosely Bernoulli.}
As a consequence we show that a large class of isometric extensions of the
standard example are conjugate to it.
