Models of growing networks are a central topic in network science. In these models, vertices are usually labeled by their arrival time, distinguishing even those node pairs whose structural roles are identical. In contrast, unlabeled networks encode only structure, so unlabeled growth rules must be defined in terms of structurally distinguishable outcomes; network symmetries therefore play a key role in unlabeled growth dynamics. Here, we introduce and study models of growing unlabeled trees, defined in analogy to widely-studied labeled growth models such as uniform and preferential attachment. We develop a theoretical formalism to analyze these trees via tracking their leaf-based statistics. We find that while many characteristics of labeled network growth are retained, numerous critical differences arise, caused primarily by symmetries among leaves in common neighborhoods. In particular, degree heterogeneity is enhanced, with the strength of this enhancement depending on details of growth dynamics: mild enhancement for uniform attachment, and extreme enhancement for preferential attachment. These results and the developed analytical formalism may be of interest beyond the setting of growing unlabeled trees.