A note on high-dimensional discrepancy of subtrees

For a tree $T$ and a function $f\text{\hspace{0.17em}⁣}:E(T)\to {\mathbb{S}}^d$, the imbalance of a subtree $T^{\mathrm{\prime }}\subseteq T$ is given by $\mathrm{\mid }{\sum }_{e\in E(T^{\mathrm{\prime }})}f(e)\mathrm{\mid }$. The $d$-dimensional discrepancy of the tree $T$ is the minimum, over all functions $f$ as above, of the maximum imbalance of a subtree of $T$. We prove tight asymptotic bounds for the discrepancy of a tree $T$, confirming a conjecture of Krishna, Michaeli, Sarantis, Wang and Wang. We also settle a related conjecture on oriented discrepancy of subtrees by the same authors.