We consider the asymptotic enumeration of labelled acyclic digraphs (DAGs) with the additional restriction of being bipartite. The analysis leads us to a meromorphic generating function in two variables for the number of bicoloured labelled DAGs whose analysis falls within the scope of analytic combinatorics in several variables. This allows us to obtain asymptotic formulas for the total number of labelled bipartite DAGs with a given number of vertices as well as for the number of such DAGs with a given bipartition (i.e., with prescribed sizes of the two partite sets).

Asymptotic expansions for the Taylor coefficients of the Lagrangean form phi(z)=zf(phi(z)) are examined with a focus on the calculations of the asymptotic coefficients. The expansions are simple and useful, and we discuss their use in some enumerating sequences in trees, lattice paths and planar maps.

A total-weighting of a graph $G=(V,E)$ is a mapping $f$ which assigns to each element $y\in V\cup E$ a real number $f(y)$ as the weight of $y$. A total-weighting $f$ of $G$ is proper if the coloring $\phi_{f}$ of the vertices of $G$ defined as $\phi_{f}(v)=f(v)+\sum_{e\in E(v)}f(e)$ is a proper coloring of $G$, i.e., $\phi_{f}(v)\ne\phi_{f}(u)$ for any edge $uv$, where $E(v)$ is the set of edges of $G$ incident to $v$. For positive integers $k$ and $k'$, a graph $G$ is called $(k,k')$-total-weight-choosable if whenever each vertex $v$ is given $k$ permissible weights and each edge $e$ is given $k'$ permissible weights, there is a proper total-weighting $f$ of $G$ which uses only permissible weights on each element $y\in V\cup E$. It is known that every tree is (2,2)-total-weight-choosable and every tree other than $K_2$ is (1,3)-total-weight-choosable. However, the problem of determining which trees are (1,2)-total-weight-choosable remained open. This paper solves this problem and characterizes all (1,2)-total-weight-choosable trees. Based on this characterization, we give an algorithm that determines in linear time whether a given tree is (1,2)-total-weight-choosable.