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This song is listed everywhere on the internet as an Irish song, but I think that association came from when the Fureys did the song in the 70s. Its almost certainly American, and it could even be a Harry Belafonte original (1960) but it really sounds like a folk melody that he re-worked.
This song was listed in the original Book from Manlia and so it has been kept.

    Rooster, The

    by trad
    We had some [C] chickens, no eggs would they [G] lay.
    We had some [G7] chickens, no eggs would they [C] lay.
    My wife said, [C] “Honey, this isn’t funny, we’re losing [G] money.”
    No eggs would they [C] lay.

    One day a [C] rooster came into our [G] yard
    And caught them [G7] chickens right off their [C] guard.
    They’re laying [F] eggs now just like they [C] used-ter
    Ever since that [G] rooster came into our [C] yard.

    We had a [C] cow-wow, no milk would she [G] give.
    We had a [G7] cow-wow, no milk would she [C] give.
    My wife said, [C] “Honey, this isn’t [G] funny, we’re losing money.”
    No milk would she [C] give.

    One day a [C] rooster came into our [G] yard
    And caught that [G7] cow-wow right off her [C] guard.
    She’s giving [F] eggnog in milk [C] containers
    Ever since that [G] rooster came into our [C] yard.

    We had a gas pump, no gas would it give.
    We had a gas pump, no gas would it give.
    My wife said, “Honey, this isn’t funny, we’re losing money.”
    No gas would it give.

    One day that rooster came into our yard
    And caught that gas pump right off its guard.
    It’s giving shell gas just like it used-ter
    Ever since that rooster came into our yard.

    We had a gum tree, no gum would it give.
    We had a gum tree, no gum would it give.
    My wife said, “Honey, this isn’t funny, we’re losing money.”
    No gum would it give.

    One day that rooster staggered into our yard
    And caught that gum tree right off its guard.
    It’s giving chicklets just like it used-ter
    Ever since that rooster came into our yard.

    We had a rooster, no love would it give.
    We had a rooster, no love would it give.
    My wife said, “Honey, this isn’t funny, we’re losing money.”
    No love would it give.

    One day that rooster came into our yard
    And caught my rooster right off its guard.
    He’s laying hens now just like he used-ter
    Ever since that rooster came into our yard.