Why Jesus died on the cross and not by any other mean? I know why Jesus died for us? But why the cross? Why not by any other mean?
For sins, the Law normally prescribed stoning a transgressor to death.


  Then shalt thou bring forth that man or that woman [among you who serve foreign gods], which have committed that wicked thing, unto thy gates, even that man or that woman, and shalt stone them with stones, till they die.
  Deut 17:5
  
  And he that blasphemeth the name of the LORD, he shall surely be put to death, and all the congregation shall certainly stone him: as well the stranger, as he that is born in the land, when he blasphemeth the name of the LORD, shall be put to death.
  Lev 24:16


So, why the tree?


  And if a man have committed a sin worthy of death, and he be to be put to death, and thou hang him on a tree:
  Deut 21:22


The phrase "worthy of death" is found elsewhere in Scripture.


  At the mouth of two witnesses, or three witnesses, shall he that is worthy of death be put to death; but at the mouth of one witness he shall not be put to death.
  Deut 17:6
  
  But unto the damsel thou shalt do nothing; there is in the damsel no sin worthy of death: for as when a man riseth against his neighbour, and slayeth him, even so is this matter:
  Deut 22:26
  
  No, nor yet Herod: for I sent you to him; and, lo, nothing worthy of death is done unto him [Christ Jesus].
  Luke 23:15


Since "worthy of death" is found elsewhere as the reason for death, what is the point about hanging the "dead" body on the tree?  "Dead" is in quotes because typically they would have stoned the sinner to death and hung the body on the tree.  It became, however, the idea that one was first punished to the point of death, as good as dead, like Isaac climbing the mountain with Abraham, and then dying on the tree/cross itself.  In either event, one was dead on the tree/cross.

The point is the tree/cross was the ultimate symbol of full and complete rejection by both man and God.


  
  There is one reason here given which has reference to Christ. He that is hanged is accursed of God, that is, it is the highest degree of disgrace and reproach that can be done to a man, and proclaims him under the curse of God as much as any external punishment can. Those that see him thus hang between heaven and earth will conclude him abandoned of both and unworthy of either;
  -Matthew Henry-
  


Burial in the Jewish culture and indeed burial or cremation throughout the world is the acknowledgement of man's creator by nearly all of ashes to ashes, dust to dust.  But if one is hung on the tree after death where your burial was postponed, then it shows the rejection of God of not returning to your Maker and the agreement by men of that state.  They would have buried Him at night, further symbolizing the vast darkness of separation from friends, family, and God.