This paper presents the results of the authors’ investigation of a combinatorial problem arising from the study of evolutionary trees. In graph theoretic terms it can be expressed as a problem of colouring vertices of a binary tree. For a given colouring of the pendant vertices of a binary tree there is a simple algorithm for assigning colours to internal vertices minimising the number of edges of the tree whose end vertices have differing colours. This minimal number is called the length of the tree. The question posed is: For given numbers of pendant vertices of assigned colours, how many trees of a particular length can be constructed on those vertices? This question is answered in two special cases. Answers to this problem are needed to establish the distribution of lengths of evolutionary trees, by which the significance of the maximum parsimony principle for selecting evolutionary trees can be judged.