The tree claims that when Christ climbed upon him, he "had to stand firm." What doctrine of Christianity might have argued that the tree was obligated to remain upright, despite the torture of its acknowledged "Lord"?
There is no doctrine, as such, that required the tree used to make the cross upon which Jesus was crucified to remain upright.
The importance of crucifixion as the means by which Jesus Christ was killed was that this was fulfillment of Old Testament prophecy. Jesus often told the apostles that He was going to die when He reached Jerusalem, and that part of His mission was to fulfill all the scriptural writings from the past that had looked forward to the coming of the Messiah. Possibly the most telling Old Testament passage in relation to the uprightness of the cross would be what is often called the "Song of the Suffering Servant."
See, my servant will act wisely; he will be raised and lifted up and highly exalted. Just as there were many who were appalled at him-his appearance was so disfigured beyond that of any man and his form marred beyond human likeness... (Isa. 52:13-14)