Why does St. Peter use the phraseology "killed by hanging on a tree" while referring to the crucifixion of Jesus?

There is also a not-theological reason: Peter and the Apostles were presumably speaking Hebrew or Aramaic in this instance (seeing as they were Jews from Judea, speaking to a Jewish council).

The terms "cross", "crucifixion" and "crucified" all come from Latin ("crux"), as this form of execution was apparently a Roman invention (in its most well known, torturous form, which was designed to make death as slow, painful, and humiliating as possible). The closest Greek term "Stauros" applies to any pole, one used as an implement of execution or otherwise.

As noted however, the idea of "hanging someone on a tree" was much older, existing in Deuteronomy. The idea of using a familiar terminology to describe an unfamiliar one is common (hence why we have, for example, a "return" key on an electronic keyboard, as a hold over from typewriters). The Romans were essentially erecting artificial trees for execution purposes.

OP: why does St. Peter use the phraseology "killed by hanging on a tree" while referring to the crucifixion of Jesus?

The word "hanging" means this.

Outline of Biblical Usage [?] to hang up, suspend to be suspended, to hang used of one hanging on a cross used of the Law and the Prophets, is summed up or hanging on two precepts -source-

So, the word "hang" is accurate for the description. It is not about "a hanging" like in the Old West.

As others noted, it is in reference to Deut 21:23, but it is also a reference to Deut 28:66 per Irenaeus, Tertullian, and Cyprian.

And again, he indicates that He who from the beginning founded and created them, the Word, who also redeems and vivifies us in the last times, is shown as hanging on the tree, and they will not believe on Him. For he says, “And thy life shall be hanging before thine eyes, and thou wilt not believe thy life.

There are multiple references to hanging on a tree in the O.T. Dt. 21:22-23 says "If a man have committed a sin worthy of death.. and thou hang him on a tree, [take him down] that day; (for he that is hanged is accursed of God)." So hanging someone on a tree shows being cursed and, surely, contemptible.

Other references, Joshua 8:29, Joshua 10:26, Esther 2:23, show enemies being hanged on trees, and seem to also suggest being cursed or held in contempt.

This fits in perfectly both with the Gospel accounts of the crucifixion, how Jesus was humiliated and demeaned, and with prophetic references to this in the O.T. including Isaiah 53 ("He is despised and rejected of men; a man of sorrows, and acquainted with grief: and we hid as it were our faces from him; he was despised, and we esteemed him not") and Psalms 22 ("But I am a worm, and no man; a reproach of men, and despised of the people. All they that see me laugh me to scorn... the assembly of the wicked have inclosed me: they pierced my hands and my feet ... They part my garments among them, and cast lots upon my vesture.") He was also accursed by God ("My God, my God, why have You forsaken me?" in both Gospels and Psalm 22) as the Dt. reference says.

So I think the best conclusion is that when St. Pete says "Jesus, whom you had killed by hanging him on a tree," he is reminding these leaders how horribly they treated the One God "exalted with his right hand, to be Prince and Saviour."

The phrase "hanging him on a tree (wood)" (κρεμάμενος επί ξύλου, suspendentes in ligno) occurs in Acts 5:30 and 10:39 (cf. Galatians 3:13).

Callan, O.P., Acts of the Apostles p. 41 says on Acts 5:30:

Hanging him upon a tree is an allusion to Deut. xxi. 23.

Deut. 21:23:

His body shall not remain upon the tree, but shall be buried the same day: for he is accursed of God that hangeth on a tree*: and thou shalt not defile thy land, which the Lord thy God shall give thee in possession.

^{*cf. again: Galatians 3:13}

The Holy Cross is the Tree of Life. The New Adam, Jesus Christ, undid by the Holy Cross what Adam did by eating the forbidden fruit of the Tree of Knowledge of Good and Evil.