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Latin word liber comes from Proto-Indo-European *h₁lewdʰ- (to grow), and later Proto-Italic *louðeros (free)
*h₁lewdʰ- (Proto-Indo-European)
to grow
*h₁lewdʰ-er-os (Proto-Indo-European)
*louðeros (Proto-Italic)
(substantivised) child. Free.
loeber (Old Latin)
liber (Latin)
free, unrestricted.