  We prove that if a knot $K$ has a particular type of diagram then all
non-trivial surgeries on $K$ contain a coorientable taut foliation. Knots
admitting such diagrams include many two-bridge knots, many pretzel knots, many
Montesinos knots and more generally all arborescent knots defined by weighted
planar trees with more than one vertex such that $1)$ all weights have absolute
value greater than one, and $2)$ there exists a weight with absolute value
greater than two. The ideas involved in the proof can also be adapted to study
surgeries on links and as an application we show that for all surgeries $M$ on
the Borromean link it holds that $M$ is not an $L$-space if and only if $M$
contains a coorientable taut foliation.
