  Bicubic maps are in bijection with \beta(0,1)-trees. We introduce two new
ways of decomposing \beta(0,1)-trees. Using this we define an endofunction on
\beta(0,1)-trees, and thus also on bicubic maps. We show that this endofunction
is in fact an involution. As a consequence we are able to prove some surprising
results regarding the joint equidistribution of certain pairs of statistics on
trees and maps. Finally, we conjecture the number of fixed points of the
involution.
