The vertical profile of embedded trees

TL;DR

This work characterizes the vertical profiles of trees embedded in the integer lattice, providing exact product-form counts for embedded Cayley and $\mathcal{S}$-ary trees with prescribed profiles and additional refinement by edge- and vertex-types. The authors develop bijective proofs, notably a two-stage map between $\mathcal{S}$-functions and $\mathcal{S}$-trees, and extend these to general negative-abelia cases via two concatenation schemes; the results cover non-negative and general embeddings and refine counts by out-, in-, and complete-types. Beyond bijections, the paper discusses alternative approaches via functional equations with Lagrange inversion and the matrix-tree theorem, offering complementary perspectives and highlighting connections to limiting measures like the integrated superbrownian excursion (ISE). The findings yield explicit, factorized enumeration formulas that illuminate the combinatorics of embedded trees and their type distributions, with implications for probabilistic limits and universality across embedded branching structures.

Abstract

Consider a rooted binary tree with n nodes. Assign with the root the abscissa 0, and with the left (resp. right) child of a node of abscissa i the abscissa i-1 (resp. i+1). We prove that the number of binary trees of size n having exactly n_i nodes at abscissa i, for l =< i =< r (with n = sum_i n_i), is $$ \frac{n_0}{n_l n_r} {{n_{-1}+n_1} \choose {n_0-1}} \prod_{l\le i\le r \atop i\not = 0}{{n_{i-1}+n_{i+1}-1} \choose {n_i-1}}, $$ with n_{l-1}=n_{r+1}=0. The sequence (n_l, ..., n_{-1};n_0, ..., n_r) is called the vertical profile of the tree. The vertical profile of a uniform random tree of size n is known to converge, in a certain sense and after normalization, to a random mesure called the integrated superbrownian excursion, which motivates our interest in the profile. We prove similar looking formulas for other families of trees whose nodes are embedded in Z. We also refine these formulas by taking into account the number of nodes at abscissa j whose parent lies at abscissa i, and/or the number of vertices at abscissa i having a prescribed number of children at abscissa j, for all i and j. Our proofs are bijective.

Figures (17)

• Figure 1: A rooted binary tree having horizontal profile $(1,2,4,3,2)$ and vertical profile $(2,2;4,2,1,1)$.
• Figure 2: The three rooted binary trees having vertical profile $(2;2,1)$.
• Figure 3: The shapes of the rooted Cayley trees having vertical profile $(2;2,1)$.
• Figure 4: Two equivalent embedded Cayley trees (the positions of the vertices indicate their abscissas).
• Figure 5: A $\{-2,-1,0,1\}$-function on $V=\{0^ 1, \ldots, 0^5, 1^1, \ldots, 1^4, 2^1, \ldots, 2^{10}, 3^1, \ldots, 3^6\}$.

Theorems & Definitions (50)

• Theorem 1
• Theorem 2
• Theorem 3: Embedded Cayley trees
• Theorem 4: $\mathcal{S}$-ary trees
• Theorem 5: Embedded Cayley trees
• Theorem 6: $\mathcal{S}$-ary trees
• Theorem 7
• Theorem 8
• Theorem 9
• proof