Ultrametric Distance in Syntax
  Phrase structure trees have a hierarchical structure. In many subjects, most notably in Taxonomy such tree structures have been studied using ultrametrics. Here syntactical hierarchical phrase trees are subject to a similar analysis, which is much siompler as the branching structure is more readily discernible and switched. The occurence of hierarchical structure elsewhere in linguistics is mentioned. The phrase tree can be represented by a matrix and the elements of the matrix can be represented by triangles. The height at which branching occurs is not prescribed in previous syntatic models, but it is by using the ultrametric matrix. The ambiguity of which branching height to choose is resolved by postulating that branching occurs at the lowest height available. An ultrametric produces a measure of the complexity of sentences: presumably the complexity of sentence increases as a language is aquired so that this can be tested. A All ultrametric triangles are equilateral or isocles, here it is shown that X structur implies that there are no equilateral triangles. Restricting attention to simple syntax a minium ultrametric distance between lexical categories is calculatex. This ultrametric distance is shown to be different than the matrix obtasined from feaures. It is shown that the definition of c-commabnd can be replaced by an equivalent ultrametric definition. The new definition invokes a minimum distance between nodes and this is more aesthetically satisfing than previouv varieties of definitions.   From the new definition of c-command follows a new definition of government. 
Mark D. Roberts