  The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the
Catalan numbers that were introduced by Raney in his investigation of
functional composition patterns \cite{Raney}. We give a new combinatorial
interpretation for all Raney numbers in terms of planar embeddings of certain
collections of trees, a construction that recovers the usual interpretation of
the $p$-Catalan numbers in terms of $p$-ary trees via the specialization
$R_{p,1}(n) =_{p} c_n$. Our technique leads to several combinatorial identities
involving the Raney numbers and ordered partitions. We then give additional
combinatorial interpretations of specific Raney numbers, including an
identification of $R_{p^2,p}(n)$ with oriented trees whose vertices satisfy the
"source or sink property". We close with comments applying these results to the
enumeration of connected (non-elliptic) $A_2$ webs that lack an internal cycle.
