This technique is called DSU on tree or Sack. The DSU refers to Disjoint Set Union, even though you are not always using DSU data structure.
The name comes from the similarity to DSU's "small-to-large" union heuristic. Similar to DSU attaches smaller trees under larger ones, this technique merges smaller child sets into larger ones.
Other names include "heavy-light merging" or "small-to-large merging." The core idea is the same: always merge small into large, never large into small. This bounds total work logarithmically.