  Structural properties of large random maps and lambda-terms may be gleaned by
studying the limit distributions of various parameters of interest. In our work
we focus on restricted classes of maps and their counterparts in the
lambda-calculus, building on recent bijective connections between these two
domains. In such cases, parameters in maps naturally correspond to parameters
in lambda-terms and vice versa. By an interplay between lambda-terms and maps,
we obtain various combinatorial specifications which allow us to access the
distributions of pairs of related parameters such as: the number of bridges in
rooted trivalent maps and of subterms in closed linear lambda-terms, the number
of vertices of degree 1 in (1,3)-valent maps and of free variables in open
linear lambda-terms etc. To analyse asymptotically these distributions, we
introduce appropriate tools: a moment-pumping schema for differential equations
and a composition schema inspired by Bender's theorem.
