The earliest writings were letters written by apostles. However the case is very strong that the first gospels were written by close associates of those apostles like John Mark, an associate of Peter and Luke, an associate of Paul.
The use of "tree" doesn't preclude a cross which would have been made of a tree trunk.
The early Christians use of "tree" was a way of linking Jesus's death to an OT scripture:
Gal 3:13 "Christ redeemed us from the Curse of the Law by becoming a Curse for us, for it is written; "Cursed is everyone who is Hung on a Tree."
Deut 21:33, "Be sure to bury Him that same day, because anyone who is Hung on a Tree is under God's Curse."
I would add that Roman use of crucifixion on a large scale is verified by historians Tacitus and Josephus and others - example:
6,000 survivors of the revolt captured by the legions of Crassus were crucified, lining the Appian Way from Rome to Capua.[39] ^ Appian, Civil Wars, 1.120
Further, no one making up a fable for a new religion in the Roman empire, would choose to have the central figure executed in a way the Romans reserved for rebels, slaves, and other enemies of the state:
Crucifixion was used for slaves, pirates, and enemies of the state. Therefore crucifixion was considered a most shameful and disgraceful way to die. Condemned Roman citizens were usually exempt from crucifixion (like feudal nobles from hanging, dying more honorably by decapitation) except for major crimes against the state, such as high treason.
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