  We study the generalized minimum Manhattan network (GMMN) problem: given a
set $P$ of pairs of two points in the Euclidean plane $\mathbb{R}^2$, we are
required to find a minimum-length geometric network which consists of
axis-aligned segments and contains a shortest path in the $L_1$ metric (a
so-called Manhattan path) for each pair in $P$. This problem commonly
generalizes several NP-hard network design problems that admit constant-factor
approximation algorithms, such as the rectilinear Steiner arborescence (RSA)
problem, and it is open whether so does the GMMN problem.
  As a bottom-up exploration, Schnizler (2015) focused on the intersection
graphs of the rectangles defined by the pairs in $P$, and gave a
polynomial-time dynamic programming algorithm for the GMMN problem whose input
is restricted so that both the treewidth and the maximum degree of its
intersection graph are bounded by constants. In this paper, as the first
attempt to remove the degree bound, we provide a polynomial-time algorithm for
the star case, and extend it to the general tree case based on an improved
dynamic programming approach.
