The in-order traversal provides a natural correspondence between binary trees with a decreasing vertex labeling and endofunctions on a finite set. By suitably restricting the vertex labeling we arrive at a class of trees that we call Fishburn trees. We give bijections between Fishburn trees and other well-known combinatorial structures that are counted by the Fishburn numbers, and by composing these new maps we obtain simplified versions of some of the known maps. Finally, we apply this new machinery to the so called flip and sum problems on modified ascent sequences.