Let $1 \leq p < 2$. Tree-like equivalence defines an equivalence relation on $C_{0,p}(V)$ that coincides with the equivalence relation $\sim$ defined by equality of signatures. In the case $p = 1$, each equivalence class $[x]$ contains an element of minimal length, unique up to reparameterisation, called the *tree-reduced representative* of $[x]$. The proof uses the theory of tree-like paths and is developed in Hambly–Lyons (2010) for $p = 1$ and Boedihardjo et al. (2016) for $p \in (1, 2)$; both arguments hinge on showing that any path with signature equal to $\mathbf{1}$ can be factored into a tree-like excursion, which is established by constructing an explicit deformation retraction onto the tree-reduced representative.