Let $x, y \in BV([0,1]; \mathbb{R}^d)$ be continuous paths of bounded variation, and consider the signature map $\mathrm{Sig}: BV([0,1]; \mathbb{R}^d) \to T((\mathbb{R}^d))$. Then $\mathrm{Sig}(x) = \mathrm{Sig}(y)$ if and only if $x$ and $y$ are **tree-like equivalent**: there exists a continuous surjection $\phi: [0,1] \to T$ onto a tree $T$ and continuous maps $\tilde{x}, \tilde{y}: T \to \mathbb{R}^d$ such that $x = \tilde{x} \circ \phi$ and $y = \tilde{y} \circ \phi$.