  A directed polymer is allowed to branch, with configurations determined by
global energy optimization and disorder. A finite size scaling analysis in 2D
shows that, if disorder makes branching more and more favorable, a critical
transition occurs from the linear scaling regime first studied by Huse and
Henley [Phys. Rev. Lett. 54, 2708 (1985)] to a fully branched, compact one. At
criticality clear evidence is obtained that the polymer branches at all scales
with dimension ${\bar d}_c$ and roughness exponent $\zeta_c$ satisfying $({\bar
d}_c-1)/\zeta_c = 0.13\pm 0.01$, and energy fluctuation exponent $\omega_c=0.26
\pm0.02$, in terms of longitudinal distance
