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A rooted planar map is a connected graph embedded in the 2-sphere, with oneedge marked and assigned an orientation. A term of the pure lambda calculus issaid to be linear if every variable is used exactly once, normal if it containsno beta-redexes, and planar if it is linear and the use of variables moreoverfollows a deterministic stack discipline. We begin by showing that the sequencecounting normal planar lambda terms by a natural notion of size coincides withthe sequence (originally computed by Tutte) counting rooted planar maps bynumber of edges. Next, we explain how to apply the machinery of string diagramsto derive a graphical language for normal planar lambda terms, extracted fromthe semantics of linear lambda calculus in symmetric monoidal closed categoriesequipped with a linear reflexive object or a linear reflexive pair. Finally,our main result is a size-preserving bijection between rooted planar maps andnormal planar lambda terms, which we establish by explaining how Tuttedecomposition of rooted planar maps (into vertex maps, maps with an isthmicroot, and maps with a non-isthmic root) may be naturally replayed in linearlambda calculus, as certain surgeries on the string diagrams of normal planarlambda terms.

A correspondence between rooted planar maps and normal planar lambda terms