A nonadsorbing, flexible polymer (in dilute solution with a good solvent) enters a pore (of diameter D smaller than its natural size, R) only when it is sucked in by a solvent flux, J, higher than a threshold value, J(c). For linear polymers J(c) similar to kT/eta (where T is the temperature and eta the solvent viscosity). We discuss here the case of statistically branched polymers, with an average number, b, of monomers between branch points. We find that there are two regimes, "weak confinement" and "strong confinement", depending on the tube diameter. By measuring J(c) in both regimes, we should determine both the molecular weight and the number b.