Binary-tree insertion coordinates are biholomorphic

For any configuration $\mathbf{z}\in X_n(\mathbb{C})$, the two leaves $L(v)$ and $R(v)$ lie in disjoint child subtrees, so their coordinates differ and $z_v\mathrm{\ne }0$. Thus every ratio ${\zeta }_e$ and the root scale $x_A$ is defined and nonzero. Coordinate differences and quotients by the nowhere-zero functions $z_{u(e)}$ are holomorphic on $X_n(\mathbb{C})$, so ${\mathrm{\Psi }}_A$ is holomorphic.

Given $(c,x,({\zeta }_e))$ in the image, set $z_A=c$ and assign the root's internal difference $z_{t_A}=x$. Recursively, if an internal vertex $v$ has assigned rightmost-leaf coordinate $r_v$ and internal difference $z_v$, assign the rightmost leaf coordinate of its left child to $r_v+z_v$; the right child inherits $r_v$. If a child is internal, assign its difference $z_d={\zeta }_e{z}_v$ and repeat. If a child is a leaf, its coordinate has now been assigned. Every leaf receives one coordinate because the two child subtrees partition the leaves below each vertex.

The reconstruction in step 2.1 uses only additions and products of $c,x,{\zeta }_e$, so it is polynomial and hence holomorphic. For coordinates in the image, induction down the tree shows that the reconstructed rightmost leaf at every subtree is its original one and that each reconstructed left rightmost leaf differs from it by the original $z_v$; therefore it recovers every original leaf coordinate. Conversely, applying the defining differences and ratios to the reconstructed coordinates returns $(c,x,({\zeta }_e))$. Hence ${\mathrm{\Psi }}_A$ is injective, maps onto its image by definition, and has the displayed holomorphic inverse there.