  We introduce the \emph{leaf-excluded} percolation model, which corresponds to
independent bond percolation conditioned on the absence of leaves (vertices of
degree one). We study the leaf-excluded model on the square and simple-cubic
lattices via Monte Carlo simulation, using a worm-like algorithm. By studying
wrapping probabilities, we precisely estimate the critical thresholds to be
$0.355\,247\,5(8)$ (square) and $0.185\,022(3)$ (simple-cubic). Our estimates
for the thermal and magnetic exponents are consistent with those for
percolation, implying that the phase transition of the leaf-excluded model
belongs to the standard percolation universality class.
