Maybe the wood is the same wood, but how do we know that the wood came the Cross. Like I said, God left us nothing material to fight over, nothing material is equal to God. He left us only His Spirit through His Word.
Re-read my post. I did not say that the Cross was stolen by the disciples, or by anyone else, at the time of the Crucifixion.
According to the tradition of the Church, the True Cross was miraculously discovered by St. Helena during the reign of the Emperor Constantine, when St. Macarius was Patriarch of Jerusalem. This is the event commemorated around the Orthodox world on September 14 -- the Exultation of the Holy Cross. It is one of the 12 Great Feasts of the Church.
It was stolen by the Persians when they conquered Jerusalem, some centuries later. After it was recovered, it was divided into many pieces and distributed to Churches everywhere, so that it would be impossible for the Church ever to completely lose this most precious of relics.
Of all miracles that we believe as articles of faith -- walking on water, rasing the dead, turning water into wine and wine into Christ's blood, -- that would be the easiest thing even without divine assistance. Remember, that cross was of significance only to the disciples. It would stand there for years awaiting the next victim.