Labelled Incomplete Binary Trees

• Labelled incomplete binary trees are rooted, plane-embedded trees with distinct labels and at most two ordered children, forming the basis for local binary search trees.
• They are studied through recursive decompositions and generating functions, linking them to Catalan numbers and bijections with no-broken-circuit sets.
• Probabilistic models using Markov chains and Galton–Watson forests reveal detailed tree statistics and edge-labeling profiles in these structures.

A labelled incomplete binary tree is a rooted, plane-embedded binary tree in which each vertex is assigned a distinct label and at most two distinguished children (left and right), with structural and labelling constraints depending on the context—most notably as "local binary search trees" (LBS) in the combinatorial literature, and as randomly labelled objects in probabilistic models of random trees, each with precise recursive properties and enumeration formulas. These structures appear in bijections with no-broken-circuit (NBC) sets in the Linial hyperplane arrangement and play a central role in the study of refined tree statistics, Markov properties of edge-labelling profiles, and extensions to $k$-ary trees and associated hyperplane or gain graph models (Forge, 2014, Metz-Donnadieu, 26 Nov 2025).

1. Foundational Definitions

A labelled incomplete binary tree on $n$ vertices consists of:

• A rooted tree on the vertex set $\{1,2,\dots,n\}$.
• Each vertex has at most two children, designated as left and right; either child may be missing (none, left only, right only, both present).
• The tree is embedded in the plane, so the order of the children is fixed.

A specialized version of these objects, the local binary search tree (LBS), is defined with the following additional "local search" constraints (Forge, 2014):

• If a vertex $v$ has a left child $u$, then $v > u$.
• If a vertex $v$ has a right child $w$, then $v < w$.
• No condition is imposed globally: the labels in the left or right subtree of $v$ are unconstrained, beyond the immediate local property.
• Any child slot (left or right) may be vacant. All four configurations (none, left only, right only, both) are allowed.

In probabilistic models (Metz-Donnadieu, 26 Nov 2025), the vertices are assigned integer labels $\ell:V(T)\to\mathbb Z$, with root label $\ell(\rho)=0$ and increments along edges in $\{-1,0,1\}$, and the existence of children is given by independent Bernoulli distributions.

2. Enumeration and Generating Functions

The number of labelled incomplete binary trees or LBS on $[n]$ has classical and explicit enumeration results, closely linked to arrangements in algebraic combinatorics and to shifted Catalan numbers in the unlabeled, unstructured case.

Enumerative Formula for LBS

For LBS, the number $L_n$ of such trees on $[n]$ is (Forge, 2014): $$L_n =\frac{1}{n\,2^{n-1}} \sum_{k=1}^n \binom{n}{k}\,k^{n-1}$$ The ordinary generating function $G(x) = \sum_{n\ge 1} L_n x^n$ does not admit a known closed form; $L_n$ grows asymptotically like $(n/e)^{n-1}$.

Catalan Enumeration for Unconstrained Tree Shapes

The number $a_n$ of (plane) incomplete binary tree shapes with $n$ edges is given by a shifted sequence of Catalan numbers (Metz-Donnadieu, 26 Nov 2025): $a_n = C_{n+1} = \frac{1}{n+2}\binom{2n+2}{n+1}$ Their generating function $B(z)$ satisfies: $B(z) = 1 + 2z B(z) + z^2 B(z)^2$ with solution

$B(z) = \frac{1-2z - \sqrt{1-4z}}{2z^2}$

Examples: $a_0=1$, $a_1=2$, $a_2=5$, $a_3=14$, etc.

3. Connections to Hyperplane Arrangements and Gain Graphs

A key structural bijection connects LBS to the combinatorics of the Linial arrangement, a well-studied hyperplane arrangement in $\mathbb R^n$ defined by hyperplanes

$H_{i,j}:\ x_j - x_i = 1, \quad 1\le i < j \le n$

and its corresponding gain graph $\Gamma = L_n$, where edges $(i,j)$ have gain $+1$. NBC trees (spanning trees with no broken circuit, under a total edge order from a height function) of this gain graph are in weight-preserving bijection with LBS (Forge, 2014).

Bijection Summary

• LBS $\to$ NBC Tree: Decompose the LBS along its right spine, recursively process left-leaning subtrees (LLBS), and reassemble using Linial-gain edges; special treatment of minimal right-spine entries preserves NBC status.
• NBC Tree $\to$ LBS: Decompose at the corner (root), order neighbors, recursively convert subtrees, and concatenate them along the right-spine to recover the plane-embedded LBS, with local inequalities preserved.

This bijection preserves structural features such as the root/corner, the degree sequence, and symmetry under label complementation $i\mapsto n+1-i$ (exchanging left and right children, corresponding to $x\mapsto -x$ in arrangement coordinates).

4. Structural and Probabilistic Properties

Labelled incomplete binary trees admit both combinatorial and probabilistic analysis:

• The degree of an internal node in an LBS equals the number of incident Linial-gain edges at the corresponding vertex in the NBC tree.
• Expected statistics of leaves, internal nodes, left/right children, and degree distributions are determined by the characteristic polynomial of the Linial arrangement (see [Athanasiadis, cited in (Forge, 2014)]), though explicit closed formulas are not given in the source papers.

In random labelled models (Metz-Donnadieu, 26 Nov 2025):

• Vertices are assigned integer labels propagating via increments $\{-1,0,1\}$, producing a Markovian evolution of edge counts between adjacent label levels.
• The vertical edge profile, $\bigl(X_k^+, X_k^-\bigr)_{k\ge 1}$, where $X_k^+$ counts edges from label $k-1$ to $k$, and $X_k^-$ from $k$ to $k-1$, forms a time-homogeneous Markov chain. When conditioned appropriately, this chain is described by explicit product-form transition kernels and yields classical enumeration results upon summation.
• A recursive "excursion-forest" decomposition at any level $m$ (cutting edges bridging levels $m-1$ and $m$) produces a two-type Galton–Watson forest structure, crucial for both combinatorial and stochastic analysis.

Markov Transition Kernel (Binary Tree Case)

For states $(p, q)$ at level $m$ and transitions $(r, s)$ at $m+1$,

$$\mathbf P_{p,q}^{r,s} = \frac{p\,4^{-(p+s)}\,(p+s) \binom{p+s}{r} \binom{p+s}{q} f_r(s)}{f_p(q)}$$

where $f_r(s)$ is a function defined in terms of sums over marked right-child and skeleton statistics, and the law of excursions is explicit via Galton–Watson processes.

5. Illustrative Examples and Special Cases

For $n=3$ elements, the exact enumeration and structure of LBS are as follows (Forge, 2014):

Example	Root	Left Child	Right Child	Additional Structure	NBC edges
(a)	2	1	3	Plane: root 2 with 1 (L), 3 (R)	(1,2), (2,3)
(b)	3	1	—	1 has 2 as right child	(1,3), (1,2)
(c)	1	—	3	3 has 2 as left child	(1,3), (2,3)

Each configuration corresponds precisely to the NBC trees of the Linial gain graph on three vertices.

6. Generalizations and $k$-ary Analogues

Two further classes extend the incomplete binary tree to $k$-ary settings (Forge, 2014):

• Local $k$-ary search tree (LkS): Rooted plane trees, each node with $\le k$ ordered children; parent label $>$ label in child-slot 1 and $<$ label in slot $k$; no restriction on interior child slots.
• Semi-local $k$-ary search tree (SLkS): As above, except parent label $>$ children in slots $1$ through $k-1$, and $<$ label in slot $k$.

These generalizations correspond bijectively, via analogous decomposition and recursive attachment rules, to NBC sets of certain gain graphs (e.g., $K_{1,k}$ and $K_{-k+1,k+1}$), with the $k=2$ case recovering the standard LBS definition.

7. Related Developments and Research Directions

Labelled incomplete binary trees, through their links to hyperplane arrangements, gain graphs, and random tree models, form a nexus across combinatorics, random processes, and algebraic geometry. Current research investigates:

• Product-form Markov chains for profile processes in labelled trees, with explicit transition kernels and connections to excursion theory and super-Brownian motion (Metz-Donnadieu, 26 Nov 2025).
• Enumeration refinements based on vertical profiles, enabling pathwise enumeration via kernels.
• Extraction of structural statistics, such as degree distributions and left-right child asymmetry, from arrangement polynomials and associated graph-theoretic properties.

Ongoing work generalizes these structures to broader classes of trees and arrangements, enhancing their combinatorial and probabilistic understanding and facilitating bijective and analytic enumeration in discrete mathematics.