Weighted Bi-Colored Plane Trees

• Weighted bi-colored plane trees are finite planar trees with alternating black and white vertices and positive edge weights, serving as critical structures in combinatorics and graph theory.
• They are enumerated using generating functions and explicit formulae that connect classical sequences like Catalan numbers with inclusion–exclusion methods.
• Their applications extend to spectral graph theory, extremal polynomial studies, and statistical models such as the two-color Potts model.

A weighted bi-colored plane tree is a finite plane (embedded) tree with an alternating vertex coloring (black and white), such that every edge connects vertices of opposite color and each edge carries a positive weight, typically an integer but sometimes taken from a larger ring when generalizing. These objects play central roles in enumerative combinatorics, spectral graph theory, Galois theory of dessins d’enfants, and the study of extremal polynomials subject to prescribed root multiplicities. Enumeration of such trees involves intricate algebraic and combinatorial constructions, including generating functions, inclusion–exclusion relations on partition posets, and deep connections with classical special numbers (e.g., Catalan, Schröder, and Stirling). This article covers key definitions, enumerative frameworks, combinatorial and algebraic methodologies, asymptotics, and connections to number theory and algebraic geometry.

1. Structural Definitions and Notation

A weighted bi-colored plane tree (WBP-tree) is a rooted planar tree with:

• A vertex bipartition $V = V^+ \sqcup V^-$ into black ($V^+$) and white ($V^-$), with every edge connecting a black to a white vertex.
• Each edge $e$ has a positive integer weight $w_e$.
• The degree of a vertex is the sum of the weights of its incident edges.
• The total weight $W$ is $\sum_{e} w_e$.
• The number of edges $E$.

Trees may be rooted by selecting a distinguished, directed edge from a black to a white vertex. The joint degree sequence (black: $\{\alpha_i\}$, white: $\{\beta_j\}$) is called the passport $\pi = (\{\alpha_i\}, \{\beta_j\})$ (Zvonkin, 2014, Pakovich et al., 2013).

Types of bi-colored weighted trees are often specified by the sequence of white and black vertex weights, or more generally, a "passport" or "type." The set of all trees with given vertex-weight sequences forms a class denoted $(k_1,...,k_s \mid l_1,...,l_t)$, where $k_i$ ($l_j$) are the weights of white (black) vertices (Kochetkov, 2013).

2. Enumeration via Generating Functions and Explicit Formulae

Enumeration depends on both the number of edges and the total weight. Define $a_{W,E}$ as the number of rooted weighted bi-colored plane trees of total weight $W$ with $E$ edges. The ordinary bivariate generating function is

$$G(x, z) = \sum_{W \geq 0} \sum_{E \geq 0} a_{W,E} x^W z^E.$$

A fundamental decomposition based on the weight $i$ of the root-edge yields the functional equation:

$$G(x, z) = 1 + z \left(\frac{x}{1-x}\right)[G(x, z)]^2,$$

which resolves to

$$G(x, z) = \frac{1 - x - \sqrt{1 - (2+4z)x + (1+4z)x^2}}{2 z x}.$$

The explicit formula for enumeration is

$$a_{W,E} = \binom{W-1}{E-1}\,\operatorname{Cat}_E = \binom{W-1}{E-1} \frac{1}{E+1}\binom{2E}{E},$$

valid for $W \geq E \geq 0$, with $a_{0,0} = 1$. This count is interpreted as: first choose a topological plane tree with $E$ edges (Catalan structure), then distribute weight $W$ among edges (stars-and-bars) (Zvonkin, 2014).

Further, a univariate recurrence for $a_W = \sum_E a_{W,E}$ is

$$a_0 = 1,\, a_1 = 1,\, a_{n+1} = a_n + \sum_{k=0}^n a_k a_{n-k}.$$

3. Type- and Passport-Specific Enumeration

The enumeration of trees with prescribed sequences of vertex weights (types or passports) requires more refined machinery (Kochetkov, 2013, Lu et al., 12 Jan 2026). Given white/black weights $(k_1,...,k_s \mid l_1,...,l_t)$ summing to $n$, "simple" (weights pairwise distinct) non-decomposable types have cardinality $(s+t-2)!$ (Kochetkov, 2013). For decomposable types, inclusion–exclusion over partitions yields:

$$|E| = \sum_{E = E_1 \cup \cdots \cup E_n} (-1)^{n-1} (v(E)-1)^{n-2} \prod_{m=1}^n (v(E_m)-1)!,$$

where $v(E) = s + t$.

For non-simple cases (repetition of some weights), counts are adjusted using symmetry factors and, if necessary, automorphism corrections (Möbius-inversion). In the context of labeled WBP-trees with full passports, the enumeration formula (using the partition lattice and Stirling numbers of the second kind) is (Lu et al., 12 Jan 2026):

$|\Tree(\Pi)| = \sum_{p \in P(\Pi)} (-1)^{|p|-1} (N-1)^{|p|-2} X(p),$

with $X(p) = \prod_{i=1}^{|p|} (|S_i|-1)!$ for each partition $p$ of the set of labels $S$ of cardinality $N$.

4. Combinatorial and Algebraic Interpretations

Weighted bi-colored plane trees admit several combinatorial encodings:

• Dyck Words: A rooted WBP-tree corresponds to a Dyck word; edge-weights are represented by paired symbols $(x_i, y_i)$ (Zvonkin, 2014).
• Balls-and-Bins: The explicit count arises from distributing $W$ indistinguishable balls (weight) into $E$ bins (edges).
• Partition Lattice: More refined inclusion–exclusion combinatorics over the refinement poset of partitions underlie type-based enumerations (Lu et al., 12 Jan 2026).
• Anti-Vandermonde Systems: For type-classified enumeration, solutions to certain anti-Vandermonde systems correspond to embedded trees; Bézout-style arguments give enumerative formulae (Kochetkov, 2013).

Weighted bi-colored plane trees are also in bijection with certain classes of permutation objects: for full labeled passports, constructive bijections exist between permutations of $N$ objects and twice-marked LWBP-trees, with explicit geometric and algebraic encodings (Lu et al., 12 Jan 2026).

5. Connections to Extremal Polynomials and Galois Theory

A weighted bi-colored plane tree of total weight $n$ and passport $(\alpha, \beta)$ corresponds to a pair of coprime polynomials $A(x), B(x)$ of degree $n$ (with identical leading coefficients), such that the multiplicity of $a_i$ as a root of $A$ (resp. $b_j$ for $B$) is $\alpha_i$ (resp. $\beta_j$). The degree of the difference $R(x) = A(x) - B(x)$ attains a minimum given by Davenport–Zannier bounds:

$$\deg R_{\min} = \begin{cases} (n+1)-(p+q) & \text{if } p+q \le n/d + 1,\ (d-1)n/d & \text{otherwise}, \end{cases}$$

where $d$ is the greatest common divisor of all parts (Pakovich et al., 2013). Uniquely determined trees by passport (“unitrees”) yield pairs $A, B$ defined over $\mathbb{Q}$, giving deep links to the inverse Galois problem and Belyi maps.

A table of unitree families classified by combinatorial schema, diameter, and rigidity is provided in (Pakovich et al., 2013). Various combinatorial "Galois invariants"—passport, automorphism group, unique vertex/"bachelor," or special monodromy group—can force a tree to be defined over $\mathbb{Q}$ even without full uniqueness.

6. Applications to Weighted Colored Maps and Statistical Models

A related, but distinct, paradigm weights edges by color multiplicity (e.g., monochromatic edges in a $2$-colored enumeration). In the 2-color Potts model on random planar maps, weighted bi-colored plane trees encode map colorings, with tree weights given by $\nu^{\text{number of monochromatic edges}}$ (Bousquet-Mélou, 2020). The generating function

$T(x, \nu) = \sum_T x^{e(T)} \nu^{m(T)}$

satisfies a quadratic equation:

$$T(x, \nu) = \frac{1 - \sqrt{1 - 4(\nu+1)x}}{2(\nu+1)x},$$

yielding coefficient formula

$$[x^n] T(x, \nu) = \frac{(\nu+1)^n}{n+1}\binom{2n}{n}.$$

Special cases recover classical Catalan numbers and Schröder numbers; probabilistic interpretations connect to Ising and Potts models on random trees.

7. Asymptotics and Special Cases

Key asymptotic behaviors include:

• For fixed number of edges $E$ and total weight $W \to \infty$, $$a_{W,E} \sim \operatorname{Cat}_E \frac{W^{E-1}}{(E-1)!}$$.
• Summing over all $E$, $a_W \sim \frac{1}{2}\sqrt{5/\pi}5^W W^{-3/2}$, reflecting a square-root singularity at $x=1/5$ in generating functions (Zvonkin, 2014).
• For weighted color models, $$[x^n] T(x, \nu) \sim (4(\nu+1))^n n^{-3/2} / \sqrt{\pi}$$ (Bousquet-Mélou, 2020).

Special and limiting cases yield known objects: proper $2$-colorings correspond to the classical Catalan numbers, while trees with edge-weights in $\{1,2\}$ and constrained total degree relate to large Schröder numbers (Kochetkov, 2013, Zvonkin, 2014).

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References:

• Enumeration of Weighted Plane Trees (Zvonkin, 2014)
• Enumeration of one class of plane weighted trees (Kochetkov, 2013)
• Enumeration of weighted plane trees by a permutation model (Lu et al., 12 Jan 2026)
• Minimum Degree of the Difference of Two Polynomials over Q, and Weighted Plane Trees (Pakovich et al., 2013)
• Counting planar maps, coloured or uncoloured (Bousquet-Mélou, 2020)