  We introduce the notion of doubly rooted plane trees and give a decomposition
of these trees, called the butterfly decomposition which turns out to have many
applications. From the butterfly decomposition we obtain a one-to-one
correspondence between doubly rooted plane trees and free Dyck paths, which
implies a simple derivation of a relation between the Catalan numbers and the
central binomial coefficients. We also establish a one-to-one correspondence
between leaf-colored doubly rooted plane trees and free Schr\"oder paths. The
classical Chung-Feller theorem on free Dyck paths and some generalizations and
variations with respect to Dyck paths and Schr\"oder paths with flaws turn out
to be immediate consequences of the butterfly decomposition and the preorder
traversal of plane trees. We obtain two involutions on free Dyck paths and free
Schr\"oder paths, leading to two combinatorial identities. We also use the
butterfly decomposition to give a combinatorial treatment of the generating
function for the number of chains in plane trees due to Klazar. We further
study the average size of chains in plane trees with $n$ edges and show that
this number asymptotically tends to ${n+9 \over 6}$.
