Measuring similarity between two mixture trees using mixture distance metric and algorithms

Ancestral mixture model, proposed by Chen and Lindsay (2006), is an important model to build a hierarchical tree from high dimensional binary sequences. Mixture trees created from ancestral mixture models involve in the inferred evolutionary relationships among various biological species. Moreover, it contains the information of time when the species mutates. Tree comparison metric, an essential issue in bioinformatics, is to measure the similarity between trees. However, to our knowledge, the approach to the comparison between two mixture trees is still under development. In this paper, we propose a new metric, named mixture distance metric, to measure the similarity of two mixture trees. It uniquely considers the factor of evolutionary times between trees. In addition, we also further develop two algorithms to compute the mixture distance between two mixture trees. One requires O(n^2) and the other requires O(nh) computation time with O(n) preprocessing time, where n denotes the number of leaves in the two mixture trees, and h denotes the minimum height of these two trees.