  Popular matchings have been intensively studied recently as a relaxed concept
of stable matchings. By applying the concept of popular matchings to branchings
in directed graphs, Kavitha et al.\ (2020) introduced popular branchings. In a
directed graph $G=(V_G,E_G)$, each vertex has preferences over its incoming
edges. For branchings $B_1$ and $B_2$ in $G$, a vertex $v\in V_G$ prefers $B_1$
to $B_2$ if $v$ prefers its incoming edge of $B_1$ to that of $B_2$, where
having an arbitrary incoming edge is preferred to having none, and $B_1$ is
more popular than $B_2$ if the number of vertices that prefer $B_1$ is greater
than the number of vertices that prefer $B_2$. A branching $B$ is called a
popular branching if there is no branching more popular than $B$. Kavitha et
al. (2020) proposed an algorithm for finding a popular branching when the
preferences of each vertex are given by a strict partial order. The validity of
this algorithm is proved by utilizing classical theorems on the duality of
weighted arborescences. In this paper, we generalize popular branchings to
weighted popular branchings in vertex-weighted directed graphs in the same
manner as weighted popular matchings by Mestre (2014). We give an algorithm for
finding a weighted popular branching, which extends the algorithm of Kavitha et
al., when the preferences of each vertex are given by a total preorder and the
weights satisfy certain conditions. Our algorithm includes elaborated
procedures resulting from the vertex-weights, and its validity is proved by
extending the argument of the duality of weighted arborescences.
