Copying with selection resulting in adaptive fitness are the hallmarks of Darwinism. However this was not a form of Darwinism based on DNA genes. The example holds up even if all the tree buds involved are precisely identical in terms of their DNA.
A kind of Darwinism of branch tips can be useful in explaining a wide range of tree-shaped structures in nature. In the organic realm, there are branches, roots, corals, circulatory systems, respiratory systems, and branching axons and dentrites. Inorganic tree-shaped systems include electrical discharges, propagating cracks, crystal growth, and drainage basins.
Many models of these types of inorganic system take a functional approach to them - for example, saying that drainage basins form efficient structures for rapidly removing water from landscapes - of that they maximize the production of entropy. From an evolutionary perspective, such models are all very well, but they are all to do with adaptive function. Adaptation is part of evolutionary theory - but it also has another side: path dependence, or historical contingency. Evolutionary theory provides a rationale for adopting a functional perspective in the first place, and it also helps to explain cases where there are deviations from what strict functionalism might predict.
While branch tip evolution is an excellent and important model of many physical systems, it has limitations. In real trees, there is cell reproduction within the branches as well as at the branch tips. Also, there are other cases where branch dynamics are important - i.e. when not all the action in the system is taking place at the branch tips. Finally, real branch tips can sometimes shrink as well as grow. In such cases, the analogy between the real tree and a family tree starts to break down. More sophisticated models involving graph evolution may be a better fit for such cases.