All three verses you reference in Acts use the same word for tree in Greek (xulon). The word is derived from the root meaning timber; so by implication anything made of wood (stick, club, tree or other wooded articles).
The reason for the translators to choose the word “tree” and not “timber” or “wooden cross”, etc was due to the connection to the book of Deuteronomy.
Deuteronomy 21: 22-23
22 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: 23 His body shall not remain all night upon the tree, but thou shalt in any wise bury him that day; (for he that is hanged is accursed of God;) that thy land be not defiled, which the Lord thy God giveth thee for an inheritance.
The significance here is that Christ became all the curses outlined in the Book of Deuteronomy “in place of” or as a substitution for all of humanity. Deuteronomy 28 tells us that Israel had to “observe and do ALL his commandments and statutes” in order to receive the blessing. Obviously, Israel (a metaphor for all humans) could not do “everything” written in the Law of Moses so Israel (ie humanity) should have received the curses of the Book of Deuteronomy. However, because of God’s love and grace through the sacrifice of Christ, humanity received the blessing.
So the truth of Christ becoming a curse for all of humanity is referenced in the New Testament in verses like the ones you reference in Acts and is more fully explained in Galatians 3:13.
13 Christ hath redeemed us from the curse of the law, being made a curse for us: for it is written, Cursed is every one that hangeth on a tree: