  We present a direct bijection between planar 3-connected triangulations and
bridgeless planar maps, which were first enumerated by Tutte (1962) and Walsh
and Lehman (1975) respectively. Previously known bijections by Wormald (1980)
and Fusy (2010) are all defined recursively. Our direct bijection passes by a
new class of combinatorial objects called "sticky trees". We also present
bijections between sticky trees, intervals in the Tamari lattices and closed
flows on forests. With our bijections, we recover several known enumerative
results about these objects. We thus show that sticky trees can serve as a
nexus of bijective links among all these equi-enumerated objects.
