  We show that the normal form of the Taylor expansion of a $\lambda$-term is
isomorphic to its B\"ohm tree, improving Ehrhard and Regnier's original proof
along three independent directions. First, we simplify the final step of the
proof by following the left reduction strategy directly in the resource
calculus, avoiding to introduce an abstract machine ad hoc. We also introduce a
groupoid of permutations of copies of arguments in a rigid variant of the
resource calculus, and relate the coefficients of Taylor expansion with this
structure, while Ehrhard and Regnier worked with groups of permutations of
occurrences of variables. Finally, we extend all the results to a
nondeterministic setting: by contrast with previous attempts, we show that the
uniformity property that was crucial in Ehrhard and Regnier's approach can be
preserved in this setting.
