The blood clotting enzyme thrombin converts fibrinogen molecules into fibrin monomers which polymerize to form a fibrous three-dimensional gel. The concentration of thrombin affects the architecture of the resulting gel, in particular, a higher concentration of thrombin produces a gel with more branch points per unit volume and with shorter fiber segments between branch points. We propose a mechanism by which fibrin branching can occur and show that this mechanism can lead to dependence of the gel's structure (at the time of gelation) on the rate at which monomer is supplied. A higher rate of monomer supply leads to a gel with a higher branch concentration and with shorter fiber segments between branch points. The origin of this dependence is explained.