Simulations: Encoding processes of simply generated trees

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In a rooted plane (ordered) tree, list the vertices v₁, …, v_n in depth-first order. The Łukasiewicz path is the walk (W_j)_{0 ≤ j ≤ n} with W₀=0 and increments W_j−W_j−1 = (number of children of v_j) − 1; it stays nonnegative up to time n−1 and ends at −1, and it encodes the whole tree. The height process is (H_j)_{1 ≤ j ≤ n} where H_j is the depth (graph distance from the root) of v_j. The (discrete) looptree of a plane tree is the planar graph on the same vertex set obtained by, for each vertex with k children, connecting consecutive children by edges and also connecting the parent to its first and last child (and discarding the original tree edges).

The following illustrations display a randomly generated tree in the upper left with the associated looptree in the upper right corner, the encoding Łukasiewicz path in the lower left corner, and the height process in the lower right corner.

These pictures were published in the following paper:
B. Stufler, On the maximal offspring in a subcritical branching process, Electronic Journal of Probability (2020), Vol. 25, paper no. 104, 1–62.