  We establish combinatorial interpretations of several identities for the
Catalan and Fine numbers and, along the way, we present some new bijections of
independent interest. Briefly, we show that C_{n} = 1/(n+1) Sum_{k}
(n+1)choose(2k+1) (n+k)choose(k) counts ordered trees on n edges by number of
interior vertices adjacent to a leaf, and C_{n} = 2/(n+1) Sum_{k}
(n+1)choose(k+2) (n-2)choose(k) counts Dyck n-paths by number of long interior
inclines. We also give an analogue for the Fine numbers of Touchard's Catalan
number identity.
