Ever since Darwin, biologists have strived to achieve the goal of building a taxonomy of living things that reflects their true phylogenetic relationships. It is often assumed that the historically prevalent notion of a tree-like structure of evolution has some real content. Indeed, resolving phylogenetic trees into perfectly dichotomous branching patterns is a general purpose in phylogenetics. In the end, it should be possible to assemble all the pieces into a grand “Tree of Life”. However, two kinds of obstacles make this an unattainable dream. A methodological constraint is that a tree-like structure is forced by the methods of phylogenetic inference on the data. Thus bifurcations are an imposition of method, not necessarily a reality Eventually, it should be clear that phylogenetic trees are not real structures, but almost certainly flawed, statistically reasonable reconstructions of historical events. The second obstacle is more important because it deals with the nature of data. It is impossible to fit all organisms into cladograms because a large fraction of life’s diversity is simply not dichotomous. The problem posed by the conflict between an idealistic tree-like goal and most of real phylogenetic data structure stems largely from focusing only on a single dimension of life –the diversity of organisms. A new vision of the Biosphere’s phylogenetic structure is presented, which includes dynamic and temporal dimensions, and may constitute a solution to the paradox posed by the notion of progress in evolution.