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Term rewrite systems are useful in many areas of computer science. Two especially important areas are decision procedures for the word problem of some algebraic systems and rule-based programming. One of the most studied properties of rewrite systems is confluence, and one of the primary benefits of having a confluent rewrite system is that the system also has uniqueness of normal forms. However, uniqueness of normal forms is an interesting property in its own right and well studied. Also, confluence can be too strong a requirement for some applications. In this paper, we study the decidability of uniqueness of normal forms. Uniqueness of normal forms is decidable for ground rewrite systems, but is undecidable in general. This paper shows that the uniqueness of normal forms problem is decidable for the class of linear shallow term rewrite systems, and gives a decision procedure that is polynomial as long as the arities of the function symbols are bounded or the signature is fixed. [ABSTRACT FROM AUTHOR]