We present the results of a search for companions to young brown dwarfs in the Taurus and Chamaeleon I star forming regions ( \tau \sim 1 and 2–3 Myr ) . We have used the Wide Field Planetary Camera 2 on board the Hubble Space Telescope to obtain F791W and F850LP images of 47 members of these regions that have spectral types of M6–L0 ( M \sim 0.01 –0.1 M _ { \odot } ) . An additional late-type member of Taurus , FU Tau ( M7.25+M9.25 ) , was also observed with adaptive optics at Keck Observatory . We have applied PSF subtraction to the primaries and have searched the resulting images for objects that have colors and magnitudes that are indicative of young low-mass objects . Through this process , we have identified promising candidate companions to 2MASS J04414489+2301513 ( \rho = 0 \farcs 105 /15 AU ) , 2MASS J04221332+1934392 ( \rho = 0 \farcs 05 /7 AU ) , and ISO 217 ( \rho = 0 \farcs 03 /5 AU ) . We reported the discovery of the first candidate in a previous study , showing that it has a similar proper motion as the primary through a comparison of astrometry measured with WFPC2 and Gemini adaptive optics . We have collected an additional epoch of data with Gemini that further supports that result . By combining our survey with previous high-resolution imaging in Taurus , Chamaeleon I , and Upper Sco ( \tau \sim 10 Myr ) , we measure binary fractions of 14/93 = 0.15 ^ { +0.05 } _ { -0.03 } for M4–M6 ( M \sim 0.1 –0.3 M _ { \odot } ) and 4/108 = 0.04 ^ { +0.03 } _ { -0.01 } for > M6 ( M \lesssim 0.1 M _ { \odot } ) at separations of > 10 AU . Given the youth and low density of these three regions , the lower binary fraction at later types is probably primordial rather than due to dynamical interactions among association members . The widest low-mass binaries ( > 100 AU ) also appear to be more common in Taurus and Chamaeleon I than in the field , which suggests that the widest low-mass binaries are disrupted by dynamical interactions at > 10 Myr , or that field brown dwarfs have been born predominantly in denser clusters where wide systems are disrupted or inhibited from forming .