A novel type of permutation tests for dendrogram data is studied with respect to two types of metrics for measuring the difference between dendrograms. First, the Frobenius norm is used, and we prove the consistency and efficiency of the permutation tests. Next, the geodesic distance on a dendrogram space is used. The uniqueness of the geodesics on every dendrogram space is proved and some existing algorithms for computing geodesics are applied. Mental lexicons of English words are analyzed as an application example of the proposed permutation tests. The difference of mental lexicons between native and non-native English speakers is examined by analyzing sorting task data that used English words taken from various word classes.