On the Enumeration of all Unique Paths of Recombining Trinomial Trees

TL;DR

The paper tackles the combinatorial explosion in enumeration of all paths on a depth-$D$ recombining trinomial tree by exploiting translational invariance and a bijection to weak compositions. It introduces a recursion-free, seed-and-march generator that outputs exactly one representative per path-equivalence class, and then augments positive representatives with a flip mechanism to generate all paths, including negative excursions, while preserving endpoint constraints. The authors derive a closed-form, entropy-based upper bound $T(D)=O\left(D^{1/2}\gamma^{D}\right)$ with $\gamma\approx1.61185$, yielding an exponential speedup over naive recursion and enabling deeper trees for improved numerics. They connect the framework to Motzkin paths and Narayana-type refinements, and discuss practical implications for option pricing, discrete-event control, and planning/reinforcement learning, highlighting potential for Gray-code optimizations and generalization to $n$-nomial trees. Overall, the work provides a principled, order-preserving, nonrecursive enumeration scheme together with a tight combinatorial count that guides benchmarking and enables scalable, exact-path analyses in path-dependent settings.

Abstract

Recombining trinomial trees are a workhorse for modeling discrete-event systems in option pricing, logistics, and feedback control. Because each node stores a state-dependent quantity, a depth-$D$ tree naively yields $\mathcal{O}(3^{D})$ trajectories, making exhaustive enumeration infeasible. Under time-homogeneous dynamics, however, the graph exhibits two exploitable symmetries: (i) translational invariance of nodes and (ii) a canonical bijection between admissible paths and ordered tuples encoding weak compositions. Leveraging these, we introduce a mass-shifting enumeration algorithm that slides integer "masses" through a cardinality tuple to generate exactly one representative per path-equivalence class while implicitly counting the associated weak compositions. This trims the search space by an exponential factor, enabling markedly deeper trees -- and therefore tighter numerical approximations of the underlying evolution -- to be processed in practice. We further derive an upper bound on the combinatorial counting expression that induces a theoretical lower bound on the algorithmic cost of approximately $\mathcal{O}\bigl(D^{1/2}1.612^{D}\bigr)$. This correspondence permits direct benchmarking while empirical tests, whose pseudo-code we provide, corroborate the bound, showing only a small constant overhead and substantial speedups over classical breadth-first traversal. Finally, we highlight structural links between our algorithmic/combinatorial framework and Motzkin paths with Narayana-type refinements, suggesting refined enumerative formulas and new potential analytic tools for path-dependent functionals.

Figures (11)

• Figure 1: Recombining Tree with Values
• Figure 1: $\mathscr{T}_{D}$ with Node-Labelings for $V_{D}$
• Figure 1: Example of Graphically Informed Constraints on $c_{k}$
• Figure 1: Distribution of the Concentration of Paths
• Figure 2: Recombining Tree with Sample Path

Theorems & Definitions (41)

• Remark 5.1: Set splitting
• Remark 5.2: Cardinality tuples as equivalence-class keys
• Remark 5.3: Ordering
• Remark 5.4
• Remark 5.5: Truncating $\hat{c}$
• Remark 5.6
• Remark 5.7
• Theorem 5.8: Reflection symmetry of counts
• Proof 1
• Remark 5.9: Extension of lexicographic order to mixed-sign tuples