A process for the preparation of a 1,3-butadiene and styrene copolymer comprising the following steps: a) anionically polymerizing a blend comprising from 5% by weight to 40% by weight of styrene and from 60% by weight to 95% by weight of 1,3-butadiene, with respect to the overall weight of the mixture, in the presence of at least one hydrocarbon solvent, of at least one lithium-based catalyst having the general formula LiR1 wherein R1 represents a linear or branched C3-C10 alkyl group, and of least one polar modifier; b) optionally, reacting the copolymer obtained in step (a) with at least one chain-end monomer selected from 1,3-butadiene, styrene and α-methylstyrene; c) reacting from 10% by weight to 70% by weight, preferably from 20% by weight to 50% by weight, of the lithium-terminated polymeric chains present in the copolymer obtained in step (a) or in step (b), with at least one coupling agent selected from liquid polyepoxides having at least three reactive sites capable of reacting with the carbon-lithium chain-ends; d) optionally, reacting the copolymer obtained in step (c) with at least one chain-end monomer selected from 1,3-butadiene, styrene and a-methylstyrene; e) reacting the linear polymeric chains remaining in the copolymer obtained in step (c) or in step (d), with at least one tin compound having the general formula XSn(R2)3 wherein X represents a halogen atom such as, for example, chlorine and R2 represents a linear or branched C1-C10 alkyl group.