Under US Copyright Law, once a tune is recorded and released, others may record and release their own versions without explicit permission from the writers or the publisher. The process is simple: pay for a compulsory license. The rate is preset by statute.
I recently learned that in addition to the Jesters’ very successful 1948 album, Little Songs on Big Subjects (described by me here and here), there was an almost identical album released on Vox Records by the Bachelors. I have not yet determined which album came out first.
The Bachelors album was also entitled Little Songs on Big Subjects and included all the songs on the Jesters’ album, plus one additional Zaret/Singer tune, The Poor Old World (see below for the lyrics). Other than that one tune, the lyrics are the same on both albums; the arrangements are quite different.
Listen and compare:
Traveling Broadens One by the Bachelors
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Traveling Broadens One by the Jesters
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Which do you prefer?
A bit of the history behind these songs is included in the Bachelors album liner notes: 
I had never heard any of the Bachelors’ versions. There is nothing about them on the Internet. Please comment if their arrangements are the ones you remember.

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Thanks to the Judaica Sound Archives at Florida Atlantic University for research assistance.
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The UN (I think UNESCO) issued a book of piano music for the original album.
I have a songbook. It was published in 1947 by Argosy Music (Hy Zaret’s ASCAP affiliate). I don’t know anything about a UN version.
No sure what prompted my internet search for the album but I found your link to the Jesters and this one. I downloaded the Jesters album but knew it was the wrong one for me. I have one double-sided 78 rpm with 4 songs and the original Bachelors album cover shown above. Sadly I can’t find the Bachelors album to download but at least I have the songs, thanks to your earlier post.