The standard coding input

List the nodes in preorder as $v_0,\dots ,v_{N-1}$, and define

Set $H_j=H(v_j)$ and append $H_N=0$. Interpolate linearly.

The identified standard input is the following consequence of Broutin and Marckert's joint coding theorem and their conditioned-degree lemma BroutinMarckert (Theorem 3, Lemma 11, and Section 6):

The interpolation and endpoint conventions make the floors harmless. The excursion $e$ is the standard normalized Brownian excursion. Conditional on degree counts, the tree is uniform among plane trees with those counts; this is the passage from that theorem to (eq:coding).

In particular,

Reversing every child order preserves the conditioned tree law. If $W^{\mathrm{rev}}(v)$ is the preorder walk value in the reversed tree, then also

Only tree-shape symmetry is used; the mark law need not be symmetric.