What are some techniques for "balancing" a tree beside heavy-light and centroid decomposition?
The paper "Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth" (Elberfeld, Jacobi, Tantau) gives a nice balanced tree decomposition based on tree contraction in $TC^0$:

https://eccc.weizmann.ac.il/report/2011/128/

Another nice $TC^0$-decomposition is based on so-called TSLPs in the paper "A Universal Tree Balancing Theorem" (Ganardi, Lohrey):

https://arxiv.org/pdf/1704.08705.pdf