Differential cross sections and analyzing powers of the /sup 13/C( p,t)/sup 11/C and /sup 13/C( p,/sup 3/He)/sup 11/B reactions to the ground and the first excited states were measured at E/sub p/ = 65 MeV. They were analyzed by means of distorted-wave Born approximation calculations including both one- and two-step processes of p-d-t or -/sup 3/He. Attention was paid to an isospin symmetry of the reaction process consisting of both isospin 0 and 1 channels, which were important in a comparable order. As intermediate states of /sup 12/C in the sequential process, five states of either isospin 0 or 1, which were strongly excited by the /sup 13/C( p,d)/sup 12/C reaction, were taken into account. A good fit between the data and the theoretical values was obtained, especially for the ( p,t) reactions, concerning both differential cross sections and analyzing powers. The calculations including one-step and sequential processs induced, in some cases, remarkable differences from the one-step calculation for backward angles.