This should not even be in question. Just Google and look up how the Romans crucified people. Jesus probably just carried the cross beam that his arms would be attached to. The pole was probably already set in the ground. The crossbeam would be like about a 6' 4"x4" post, thus the "cross". Easy enough for a healthy man to carry, but Jesus was beaten before crucifixion and was weak from pain, lack of sleep, and blood loss, thus needing someone to carry is cross.
Romans used two types of crucifixion, on nailing, and one using ropes to tie them the beam and pole. Using ropes caused a very slow death, usually after being beaten, with no food, water, and they were bleeding from being beaten, and took a few days to a week to die. Using nails for the hands and feet as well as the ropes, was more painful, but the person usually bled out within a day, like Jesus.
Jesus was not nailed to a tree, but to a cross or crossbeam on a pole. The crossbeam was usually attached to a large pole with a large wooden pin, so the dead person could easily be removed for the next person using another crossbeam.
Facts are facts, and we have a lot of Roman written history.