This paper presents a k-out-of-n recursive information hiding scheme based on an n-ary tree data structure. In recursive hiding of information, the user encodes additional information in the shares of the secret, intended to be originally shared, without an expansion in the size of the latter. The proposed scheme has applications in secure distributed storage and information dispersal protocols. It may be used as a steganographic channel to transmit hidden information, which may be used for authentication and verification of shares and the reconstructed secret itself.