ABSTRACT. Top trees with DAG representation can be used to compress huge ordered-tree data such as XML documents. However, one ordered tree can be represented by several top trees, so it is necessary to efficiently decide which top trees represent the same tree for higher compression rate. In this paper, we give a complete axiom system for the equational theory of top trees, called the cluster algebra. In order to prove the completeness, we introduce a reduction system on cluster algebra, and show the strong normalization and the unique normal form property.