I'd like to decide whether, given a connected graph $G = (V, E)$ and an integer $k$ as input, $G$ admits two vertex-disjoint subgraphs $T_1 = (V_1, E_1)$ and $T_2 = (V_2, E_2)$ such that $T_1$ and $T_2$ are trees, $V_1 \cup V_2 = V$ (so in some way they are spanning), and $|V_1 - V_2| \leq k$ (they aren't too different in size).
Some graphs, such as perfect binary trees or Hamiltonian graphs, do admit such balanced spanning trees ($k=o(1)$), whereas others do not, such as star graphs ($k = n - 2$).
Is this NP-hard? Is it already NP-hard when fixing $k=0$? Also, is the smallest $k$ for which such trees can be found a known parameter, like maybe ``balanced vertex-disjoint spanning tree distance'' or something? I did not find any information on vertex-disjoint spanning trees like this, I always end up with an article on edge-disjoint spanning trees (which each individually are spanning).
EDIT: vertex-arboricity (resp. equitable vertex-arboricity) seems quite close, but here one aims to partition the vertices such that their induced subgraph is a forest (resp. forests of the same size). They don't seem interested in the exact number of partitions (whereas I fix it to two) and I'm not interested in induced subgraphs or forests.