The most common and most likely answer to this crossword clue is the 4 letter word BARK
I've seen this crossword clue in The Times
B
A
R
K
More clues leading to the result BARK
Buddy explains!

The straight part
We know that "Bay/tree's part" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case bay/tree's part = bark.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does BARK mean?
BARK noun- The short, loud, explosive sound uttered by a dog, a fox, and some other animals.
- An abrupt loud vocal utterance.
- To make a short, loud, explosive noise with the vocal organs (said of animals, especially dogs).
- To make a clamor; to make importunate outcries.
- To speak sharply.
- The exterior covering of the trunk and branches of a tree.
- Peruvian bark or Jesuit's bark, the bark of the cinchona from which quinine is produced.
- Hard candy made in flat sheets, for instance out of chocolate, peanut butter, toffee or peppermint.
- The crust formed on barbecued meat that has had a rub applied to it.
- The envelopment or outer covering of anything.
- To strip the bark from; to peel.
- To abrade or rub off any outer covering from.
- To girdle.
- To cover or inclose with bark, or as with bark.
- A small sailing vessel, e.g. a pinnace or a fishing smack; a rowing boat or barge.
- A sailing vessel or boat of any kind.
- A vessel, typically with three (or more) masts, with the foremasts (or fore- and mainmasts) square-rigged, and mizzenmast schooner-rigged.