We generalize some current approaches for RNA tree alignment, which are traditionally confined to ordered rooted mappings, to also consider unordered unrooted mappings. We define the Homeomorphic Subtree Alignment problem, and present a new algorithm which applies to several modes, including global or local, ordered or unordered, and rooted or unrooted tree alignments. Our algorithm generalizes previous algorithms that either solved the problem in an asymmetric manner, or were restricted to the rooted and/or ordered cases. Focusing here on the most general unrooted unordered case, we show that our algorithm has an O(n T n S min (d T, d S)) time complexity, where n T and n S are the number of nodes and d T and d S are the maximum node degrees in the input trees T and S, respectively. This maintains (and slightly improves) the time complexity of previous, less general algorithms for the problem. Supplemental materials, source code, and web-interface for our tool are found in http://www.cs.bgu.ac.il/~negevcb/FRUUT.